A rook pawn wins only when the attacking king secures b7 (or b8) before the defender reaches c8 or the corner.
Rook pawns are drawish, but not always drawn. White's king on b6 is a single step from the magic square. 1.Kb7! wins: the king now covers a8 and b8, the black king can never get in front of the pawn, and a5-a6-a7-a8 cannot be prevented.