Yes, as they can no longer complete the current turn. Traditionally chess is played with time limits, so it makes sense that death causes the current turn, and only the current turn, player to lose.
I propose the alternative that death is instant disqualification regardless of game state, which incentivises murder as a strategy and adds an additional psychological tax but also a necessity to resolve disputes in a way that really makes the Han/Greedo distinction an important one and would make for a fun "spaghetti western involving but not necessarily for intellectuals" flick.
> spaghetti western involving but not necessarily for intellectuals
It would have be scored by Ennio Morricone, who was passionate about chess: "When I was a kid I had two ambitions, to become either a physician, or a chess player, not a musician." [0]
For practical purposes this is not correct though. It would be trivial for the person whose turn it is to make _any_ legal move to shift the current turn over to their now-deceased opponent and as such win the game.
Making a move does not necessarily shift the turn to your opponent. You can have a position where all of your possible legal moves stalemate your opponent, in which case your move immediately ends that game and is the last move of the game.
Tangentially, this reminds me of an amusing chess conundrum. Is it possible for a position to occur in a legal game of chess where all pieces and pawns are on the board, all of them are on their original squares, and white does not have the move? For purposes of this question, a knight or rook is considered to be on its original square either if it is on the square it started on or if it has swapped positions with the other knight or rook of the same color.
> Is it possible for a position to occur in a legal game of chess where all pieces and pawns are on the board, all of them are on their original squares, and white does not have the move?
No, and that's quite easy to prove: With all pawns on their original position, the only valid moves are by knights and rooks, and for each such move the moving piece changes the color of space it occupies (for rooks that's because they have only 1 empty place to move).
That is, in order to return to the initial position, each side must make an even number of moves (knights exchanged or not), so next move will be always white's.
I think there is an edge case where both people die at the same time, but one is about to make a move and drops the piece as they die, and it lands on the board. Is that move valid? Or not as they are already dead when it lands?
This is logical, but the situation when the other player dies needs some solution too. Because the living player finishes their turn and then ... what? There isn't even anyone to have a turn anymore.
So for the sake of completeness, I would expect the rule to cover all deaths at the table.