Hcfyk

Yes, this is part of the translation convention

People tend to think of translation as a word-to-word equivalency, but it isn't. Different languages have different grammars, and each language words for concepts that can't be clearly defined in other languages. Translation is about communicating meaning and intention, and wordplay can be a vital part of that.

Part of the translation convention is to assume that an English language pun is replacing a roughly equivalent pun in the native language.

You can see this in good translations of existing novels, where puns in one language will be transformed int similar, but different puns in the new language. TvTropes calls these transformations Woolseyisms (Although Woolseyisms go beyond just puns). You won't find them universally, because translating is hard, but there are some good examples there.

Yes, non-English-speaking characters can use English wordplay.

For example, none of the people in Shakespeare's Julius Caesar are really speaking English, yet there is no problem that there are puns, some meter, and even a little rhyming—all in English.

I am from Spain, so I have a lot of experience reading books in one language (Spanish) where the characters were supposedly speaking in a different language (usually English). I can give you a reader's point of view, then.

If you are writing a story in English, aimed at English speakers, where the characters are supposedly speaking their own native language but their words magically appear as English to us readers (e.g. Shakespeare's Julius Caesar, Golden's Memoirs of a Geisha and mostly any Fiction work set in a non-English-speaking place), then you have two options:

• 
  Use non-English wordplays, translate them to English as literally as possible and leave a footnote explaining the actual meaning. This can remind the reader that they are looking at a different world and culture, and make them more engaged with your story.
  
  I'll always remember when I was a teenager, reading a Spanish translation of "Hackers" by David Bischoff I came across the sentence "El semáforo se puso en verde. Verde significa dinero.", which didn't make much sense until I read the footnote explaining that US banknotes are green (Spanish banknotes back then had different colors). I liked learning that, and it reminded me that these people were not Spaniards, even if their words appeared as Spanish to me.


• 
  Use English wordplays, but cautiously and only for unimportant conversation. If you need some character to make a joke or show some wit, then cool, go ahead, let'em make a pun here and there, but be careful: if the pun uses some too-obviously-not-English elements, or if you use it for any kind of plot-advancing situation, for readers that know about your characters' culture it can break suspension of disbelief really fast.
  
  I can't remember any actual of example for this, but imagine your characters were supposedly from Spain and so they were aptly named Juan, Ana, Ignacio and Lucas. And the plot depended somehow on their initials forming the word J-A-I-L... Well, how convenient that their Spanish names formed an English word! (/sarcasm).
  
  Or maybe one of them is called "Irene" and you make a play on words with "seen"... but in Spanish "Irene" is pronounced "ee-REH-neh" not "eye-REEN".
  
  Or they are making jokes about the Easter Bunny, when that is absolutely not a thing in Spain.
  That kind of things can be picked up by your more savvy readers and kill the mood.

However, if you are writing a story with some parts in a non-English language, but still aimed at English speakers, then you only have one option:

• 
  Avoid English word plays if they don't make sense in the language you're using. As an English reader, chances are that I know the old "Why was six afraid of seven?" joke, but no matter how you translate the pun to Spanish (either "¡Porque siete ocho nueve!" or "¡Porque el siete se comió al nueve!" or even "¡Porque el siete hizo un bizcocho con el nueve!"), it's not gonna work. If your readers can read that foreign language, they will realize the joke makes no sense in that language; and if they can't read it... well, it wouldn't make a difference anyway so just don't do it. If your characters are using a different language, they must do so coherently.
  
  A bad example for this is Tolkien's Lord of the Rings, when they realize "Speak friend and enter" is a play on words ("speak" can mean both "talk" and "say"). There's just a little problem: in Spanish, "speak" can only mean "talk"; we use a different verb for "say". So when I read "Habla amigo y entra" and then Frodo went "Hey, maybe 'Habla amigo' means that we must just say 'amigo' to enter", I was like "What? No, why would it mean that? It'd say 'Di amigo' not 'Habla amigo'. What's in that hobbit pipe-weed of yours?"
  

TL;DR

• Non-English wordplays translated to English: OK
  


• English wordplays in English: Sometimes OK, careful not to include out-of-place cultural references, do not use for plot-advancing either.


• English wordplays translated to non-English: AVOID
  

Would it be jarring if in an original (non-translated) story, the characters, who don't speak English in-universe, use "untranslatable" wordplay/puns that are specific to English?

The only time I'd say "yes, it would be jarring" (as opposed to "no, don't worry about it") is when the difficulty that these characters have in speaking English is a sub-plot, running gag, etc. For them to suddenly use such language would jar the reader and other characters.

You then could then either subvert the "foreigners speak Engrish" trope or move forward the (sub-)plot by having the character explain how they learned this bit of colloquial English.

A speaker of a foreign language can create a pun, or some sort of oddly constructed phrase in the reader's language by mistake.

In Phillip K. Dick's novel, "The Man in the High Castle," a Japanese character, Mr. Tagomi, says, "Fleece-seeking cortical response." It takes another character a second to realize that Tagomi means "woolgathering."

It seems to me that one could have a foreign speaker make unintentional puns, or try to translate puns in his own language literally with humorous results. This gimmick requires an explanation and can be used too many times.

Can non-English-speaking characters use wordplay specific to English?

Assuming the question is from the perspective of a writer, rather than of a reviewer or teacher, the answer is that it's a decision for the author to make. As others mentioned, it can certainly be done and there are existing examples. The actual question about wordplay doesn't stray from the principal question very much.

Would it be jarring [to] use "untranslatable" wordplay/puns that are specific to English?

It certainly could, and this question has an answer saying so, as well as other discussion on what is awkward and not, plus general thoughts about in-story language use. When you say jarring, I come to think of Jar-Jar Binks: Characters can be found jarring also when not using English wordplay.

In the end, I think it matters more how well it fits the story than how well it translates. Puns are commonly frowned upon (a.k.a. dad jokes). Would the story work with a more general joke or a word of wisdom? Is it a re-occurring theme or a one-time thing?

The audience probably doesn't care, but I definitely do...

I seem to read the question as if you already have the answer here. Did you say you're learning towards "no pun"?

Even your pro-wordplay arguments are diminished as "poetic license" rather than "good writing". Does your storytelling benefit from this particular pun? Is it too restricted without it?

Of course, in that case the answer may as well be: There's no harm in trying the unconventional. Leave your comfort zone and do the unexpected!

I hope this answer can help writers of any inclination to let the story go beyond strict rules.

PS. The prompt when adding questions on this site says: "We prefer questions that can be answered, not just discussed." Even though it was up-voted, IMHO this question opens for giving advice rather than for finding facts. With the detailed research OP already did, an alternative, more fact based, question could have been "Are there any other factors and arguments to consider?" or "Please assist me in finding literature examples where this occurs."

One of the fun parts of learning a language is learning vocabulary, puns, turns of phrase, etc that just don't translate to your mother tongue.

Any English learner (any learner of any language) is going to listen for those special things and use them as soon as the opportunity comes up (as long as s/he feels comfortable enough to try).

Part of the fun for a native speaker who is friends with a non-native speaker is listening for the times when the latter gets these phrases just a little bit wrong. There's even an American cartoonist married to a German woman who delights in periodic strips gently poking fun at her errors (presumably with her approval).

When we learn other languages, we dutifully memorize vocabulary and grammar. But we don't really know the language until we can say things you just can't say in any other language (and until we have dreams in the new language, but that's for another question).

Your near-fluent characters will not only use the words and phrases you ask about, but likely go out of their way to learn about them and to work to master them. How they do this will vary by each speaker's personality, the natives they speak with, and context. But it will happen.

Even then, there are some jokes that are "Universal Jokes" such as the following:

Wife: Does this dress make me look pretty or ugly?
Husband: I would say pretty ugly.

The word play here is almost universal and the gag translates into nearly every language just fine and without losing meaning. It even works in language families that are not connected to English, such as Chinese families.

In specific cases, for example, Japanese and English Jokes that pun off of the English word "Dream" (Yuma in Japanese) will work fine as the two words can be used to cover the same meanings of the word Dream equally. A lot of Japanese Punning deals with numbers having multiple sounds and these sounds can sound like real words in a combination of numbers. This isn't unknown to English punning as many numbers can sound like other words as well. Suppose I title my chapter on the multiplication table of 6 "The Joys of Six (Sex)) and you'd get the idea. It's not uncommon for Japanese pop stars and companies to sign their twitters with a number combonation and it's why so many Japanese titles use the word "Go": Modern Japanese, which has a large amount of English Loan Words from post-WWII occupation, became well aware that English also use the word "Go" but instead of for a number (5 in Japanese) as an action. For example, the Japanese title for "Speed Racer" is "Mach Go Go Go" which contains a triple pun as in addition to the English Go and the Japanese number Go, the Japanese also use "Go" as "Type" in machine iterations (Similar to a Iron Man's "Mark 1" designators for his various suits). So the correct title could be read as "Mach Type 5, Go" if properly translated. The Super Sentai Series "Kyukyu Sentai GoGoV" (Provided footage for "Power Rangers: Light Speed Rescue" in the States) is an even more extended pun. Kyu is Japanese for 9 and Kyukyu (99) is also the word for Rescue. The V is being used as a Roman Numeral Five and the English word "Five" was used when announcing the name properly. The series was the first after the Sixth Ranger concept hit the screen to have an entire run without an additional sixth hero, so the name highlights the fact that it will only focus on five heroes. Now, where the pun comes in is that the Rescue themed series debued in 1999 which is punny enough, but the number 555 is also a rescue themed pun, in that it's the Japanese equivelent of 911 in the States. Again, here it can be funny if you understand the nuances of the language's popular puns, but requires a lot of explaining.

Another good series of jokes in English is Russian jokes. Aside from a few genral knowledges of Russian Culture, the humor is generally accessible in translations and works well. In fact, what is to be considered the funniest jokes in the world (The Sherlock Homes and the Stolen Tent) originated in Russia (the character is wildly popular and Between Holmes and Wattson, they can afford to set up a lot of humor at the expense of the English.).

With very few exceptions, you never want to interrupt the flow of the engaged reader, or endanger his or her suspension of disbelief. With that said, what will interrupt the flow depends on what the flow is.

If you are placing the reader inside the non-English speaking characters --if you are giving a sense of what the world feels like for them from the inside --then English puns may be fine, even if they don't make literal sense. (This also holds if your narrative is deliberately non-realist.)

If you are viewing these characters from an external perspective --if they are not the POV characters --AND this is a realist narrative, then it makes more sense to render the jokes they would tell in their own language, not the ones an English speaker would make up.