White cannot win by merely advancing the Pawn. A little preparation is needed.
A simple count shows that the Pawn can reach a7 in four moves, but that the pursuing King, rushing back along the diagonal, will head it off at b7.
The play would be: 1. a4, Ke4 2. a5, Kd5 3. a6, Kc6 4. a7, Kb7 and the pawn is lost. White must try other means:
1. Kf5!
This prevents Black from moving along the diagonal to stop the Pawn, and threatens to win by 2. a4 ... 3. a5 ... etc.
1. ... Ke3
Black seizes a new diagonal and is ready for this continuation: 2. a4, Kd4 3. a5, Kc5 4. a6, Kb6, and he wins the Pawn.
2. Ke5!
Interferes with that little idea, and once again threatens to win by pushing the Pawn.
2 ... Kd3
Now if 3. a4, Kc4 4. a5, Kb5 and Black draws.
3. Kd5!
Cuts Black off a third time from pursuing the Pawn.
3. ... Kc3
This gives White the last chance to start the Pawn prematurely. If now 4. a4, Kb4 nails the Pawn down.
4. Kc5!
This is decisive! Black's King is shut out completely. The continuation (regardless of anything Black does) will be 5 a4, ... 6 a5, ... 7 a6, ... 8 a7, ... 9 a8 (Queen).
White wins.