This is a L&D that I encountered in a Shusaku game. The white 'monkey jump' is not played in the game, as white (Ota Yuzo) resigned a little after this situation was set up. This L&D was purely set up by me because I was wondering if black was still unconditionally alive after white made this small group here.
It's an interesting situation, especially with black's surrounding stones (to the left) being in atari if they would push at J1 and if white connected at K1, meaning black would have to connect at F1 in gote, leaving white with the initiative.

I've tried several variations and I think I found the answer, but I'm not so confident I have read it correctly or in the best way. But I think black is alive.

Alternatively, maybe white has a better move here than P1 (I think P1 would be an overplay and white at N2 would be better in endgame, or what about O1?)

Again, P1 is not in the game, it's a move I was wondering about.
Also note this is not the full board, I only put the relevant stones there and stones to show groups as alive.

I don't see how Black can live after White at P1 (so I don't understand why White resigned )

White resigned because Shusaku killed off a white group on the left of the board. Looking at the final board position, I think black even wins if this group of his dies. So the L&D is not related to white resigning. Probably black could've lived for certain with another move here, but he took sente to kill a big group of white.

So forget about the resigning I am interested in the local L&D, though!

If black's life is supposed to depend on the ability to poke an eye at J1 (even in gote), is there some reason why white can't just atari at J1 first to secure the eye, and then play this monkey jump?

Nope.
Only reason I hadn't included it is because I started doing the L&D very locally. Only by making the .sgf for posting it here, I noticed the additional parameter of black being in liberty shortage there, expanding the L&D question. So actually with this fact I think black is dead. If black didn't have a liberty shortage there, I think he's alive. I'm not certain, though

If Black is not restricted by his shortage of liberties (by having a stone at F1) then he can kill White by playing J1 and as such White has no time to play the monkey jump, allowing Black to live.
If Black is restricted by this shortage of liberties, then White can live in sente by playing J1, giving him time to kill Black with the monkey jump.