Can White win this?

I can't see how Black will stop the A pawn.

a8 is not of the colour of the bishop!

Never mind. White can't win.

Show me how!

Yes, it can 1) Kb1; Kxp 2) Kc2; Kb6  3) Bh4 Kf5 4) Kd3 Ke6 5) Kc4 Kd6

6) Kb5 winning!  if 6) ...Kd6 7) Bg3+ cutting off the king's advance to the queen side; any other move by black it is followed by Kb6 and white gains control of the queening square, then it is a matter of capturing the A pawn which isn't going anywhere and pushincg the a pawn to queening!

How about this?

The first move was not obvious!

If Black refuses the bishop:

By the way, is this a game of yours, or an existing puzzle?

I don't think it's a waste of time. It is in the way of the Black king's path to the A8 square. Otherwise he must decide to go in front of it or behind it, and unless I'm wrong, White makes it to the corner either way. 

That's a draw. The black king walks to a8, if the white king tries to prevent this black will push the d-pawn and force white to go after it (freeing the way to a8 for the black king).

I think white should try to reach a position like this:

(Edit: and then white can move the king towards the action and put black in zugzwang. And it looks like white is just in time to achieve this position.)

mathot, that position is a draw with White or Black to move.  (Black can reach the a8 corner.)

You can see for yourself with any engine that has access to the 5 piece tablebase set, or the shredder tablebase website:

http://www.shredderchess.com/online-chess/online-databases/endgame-database.html

@MrEdCollins: yeah, you are right, that was a mistake.

I will not use an engine or tablebase (yet) though, it spoils the fun 

"That's a draw. The black king walks to a8, if the white king tries to prevent this black will push the d-pawn and force white to go after it (freeing the way to a8 for the black king)."

If you mean after 1. Bb2 axb2+, no, that is not a draw. The tablebase at the shredderchess site Mr. Collins links to shows it as a win for White in 16 moves. White beats Black to A8. After 2. Kxb2 Kxh5, White has 8 different moves that win by force, a4 and Kc3 being the shortest.

@JamieKowalski: I meant the final position in #8 with black to move is a draw. I hadn't noticed you already posted a solution in a later post (edit: actually that's not the solution after all, just another variation).

Forgot how to count!? it is a waste of time and puts the black king a tempo behind the white king! after Kf7, if White keep moving the king, Ke8 followed by Kd8, black beats the white King to the queening square.

If after Ke8 white plays Be7, Blacks takes the bishop and the freed Queen pawn should guarantee the draw. i don't use engines analyze the old fashioned way like Bobby used to do!

@JamieKowalski: I took another look, and in the variation you post in #8, white has to play 4.a4 to win the game. 4.Kd4? is not needed and leads to a draw because the black king will be just in time at a8 (white's king can't prevent that due to blacks pawn).

Actually 3.a4 seems even more to the point (although 3.Kc3 does not giv away the win).