Do Sign Languages have puns?

Yes! My favorite example is the ASL sign for pasteurized milk, which is just the sign for milk, but moved ‘past your eyes’.

https://youtube.com/watch?v=KGprfOaUxsA

Edit: another comment reminded me that I should specify that this is American Sign Language.

That is amazing.
Adam Hills had a couple comedy specials where he has an interpreter and some of the jokes get 2 rounds of laughter as you see how it gets signed.
- The Oatmeal author: https://youtu.be/ZZ_BtZ-5O60&t=363
  Good from start, but that's the highlight.
Ehm... I adjusted the volume to hear the joke. I know I'm normally an idiot, but I think I'd be needing some sleep.
- We’ve all been there

Yes! And they work a lot like the puns in spoken languages - you got two things that are conveyed in a similar way, but have a different meaning, so you sub one into another creating an unexpected result.

For example, in Libras (Brazilian Sign Language) I've seen once a person starting with a "I don't give a fuck" [signs: today I don't-care], and then immediately "distorting" the don't-care into a "poor thing"/"I feel sorry" - because even if the gesture is different, the hand configuration is the exact same, using the ring finger at 90° from the palm. (The other person was clearly amused.)

That's a pun, in the exact same way as starting with a spoken word, pausing, and then completing it in a way that conveys something else.

I read a book last year (Song for a Whale) about a Deaf girl who would play a game with her grandfather where they would create a story together while using the same hand shape all throughout. So maybe they would make a fist, or ann open palm, or a “y” shape and then the story was created using signs that used that hand shape. If you couldn’t continue the story with the same hand shape you lost. Not exactly a pun but I thought it was interesting.

Related question: are there sign language specific dad jokes? That is, corny jokes that work in sign language but not in spoken language?

I don't know about puns but I once saw a documentary about a deaf and mute married couple who had a baby. The interesting thing was that the baby started "babbling" with its hands. Can't remember if it was deaf and/or mute as well.

Mute.
Please do not use "dumb".
- Thanks!
  I had asked DDG for a translation and it only told me about "dumb". In the back of my head I thought that there was a better word but it wouldn't come to me.
I know historically “deaf and dumb” meant deaf and mute, but, at least in the classes I took in college, I was told we don’t use that terminology anymore, for hopefully obvious reasons.
- The euphemism treadmill spins round and round...
My non-deaf and non-mute baby also does hand babbling and will sometimes try to mimic us when we sign. Normal baby behaviour.

Depends on what you mean by puns. As someone else pointed out, some signs in and of themselves are jokes, but there are also plenty of jokes in ASL in particular that don’t translate super well, so they’re really only funny in the language.

My understanding is that sign languages are far less ambiguous than natural languages, so i suspect there would be far fewer.

Yes. Turtle.