The stalemate

Stalemate is where the player whose turn it is to move is not in check, but has no legal moves.  There have been a few in practice.  Here is one game that occurred in the British Chess Championship in 1929 (won by Mir Sultan Khan) at Ramsgate between William Fairhurst and T. Tylor.  It could have ended in either a stalemate or a perpetual check.

39...Kh8 40.Rxh7? Kg8! and there's no perpetual as the bishop can take the rook, should it check again. However, 39...Kh8 40.Rf8+ is a perpetual, so all is well that ends well

Nice example.

Hicetnunc,
You are right.  With both sides have less than a minute each, perhaps Black should play Kh8 to see what happens.  White has to play Rf8+, leading to a perpetual check and not a stalemate.