I believe that in this case the game would not be immediately be drawn after Bxb1 (or even after Kxb1).
5.2 b. The game is drawn when a position has arisen in which neither player can checkmate the opponent’s king with any series of legal moves. The game is said to end in a ‘dead position’. This immediately ends the game, provided that the move producing the position was legal. (See Article 9.6)
However some national chess organisations override this rule with one which states something to the effect that the game is drawn if there is no forced mate.
I've often seen children play on when it's something like king and bishop each, with one side short of time. If I see it in time and I am in charge then I will declare the game drawn.