You can always attempt to get stalemated which is a draw, or get a draw my threefold repetion , or in theory the 50 move draw rule as well( if say 47/48 of those already had happened etc).
I this case we were talking about situation where other player is literally dead. So I imagine draw would be granted, on part of other party not making any moves...
King+pawn vs king can only be won if the other guy makes a mistake. However, the other guy can make a mistake. The draw situations are king+knight vs king or king+bishop vs king.
The K+P vs. K scenario being discussed is not the one where the K player dies (loss, because K+P could have won). It’s the one where the K+P player dies (draw, because K could not have won).
K+N vs. K or K+B vs. K are already an automatic draw regardless of the players’ ability to make future moves, so death would not be a relevant factor there.
Pawns can get promoted, so there are are many king vs king + pawn positions that are a win for the player having the pawn.