Can Black win this game??

Recently finished this game.

Game 3540055

I offered a draw at move 36. It was refused and the following conversation took place at moves 37 and 38:

Me, move 37:
apologies as you are a better player but how is this not a draw? if it were me against a 1300 player I would not know how to make this a Black win. I've looked! How can you win if all I do is hold? Please explain, I will resign. Otherwise accept my draw offer and my salutations sir.
His response:
I have a slight point advantage since I am one pawn up on you, so I can probably worm one of my pawns across the board. I can also hope that you will make a mistake and I can claim another pawn which will give me an advantage. The main reason I am not accepting the draw is because the point standing in the tournament is so close between Rookguy and myself. If I take the draw and he beats me then I lose the tournament but if I claim a win then even if he beats me then we push to another round. If it looks like I will draw or win against him I will draw with you but I want to see what his next few moves are before I make that decision.
Me move 38:
fair enough. thanks for that. I intend to do nothing but move my Bishop to guard whichever of my pawns your King attacks I won't move King or pawns again.
His response:
I really think that without moving your pawns or king I can still get something across the board but since Rookguy accepted my draw I will accept yours by offering you one now.

Did I miss something here? Could Black have won if all White did was push Bishop around. I realise that his motives were to win tournament and fair enough I salute that but was there really a win here?

Thanks for you help.

Looks like a draw to me too, his pawns cannot pass and you can always defend your own pawns against his king as you said. well played.

Drawn.....

dead draw. He was either pretty wrong with his claim that he could still have gotten something across the board or simply bluffing.

Thank you all for replies, especially Schakulr as you are a 2000+ player.

Glad I missed nothing.

Obvious draw - doesn't take a high rating to know that.

the last pawn exchange was bad for black. that's why it is a draw.

No way black can win the final position.
White's king can't be pushed off f3 and there's no way for black to win white's a- or h-pawn.

Any time you reach an endgame where the only pieces left are "opposite colored bishops" it is usually a draw even if one side has an extra pawn.

Originally posted by Red Night
Any time you reach an endgame where the only pieces left are "opposite colored bishops" it is usually a draw even if one side has an extra pawn.

I would say that's a win for black.

Originally posted by Tigran P
I would say that's a win for black.

Originally posted by Dragon Fire
and sometimes even 2 pawns advantage is not enough to win.

Originally posted by Tigran P
I would say that's a win for black.

Originally posted by Red Night
You're absolutely right. I almost added that, but did not want to muddy the water. (At this point in the discussion, I just wanted to introduce the concept of "opposite colored bishops"😉