EITHER Side to Move DRAWS!
I know... I know... Some of you have been in this position and won as White. It's especially easy to do when White has the move, but it still requires Black to play inaccurately. We'll start the proof of this drawing evaluation assuming White to move.
White has to pick a side of the pawn to move to, because stepping backwards to c2 would immediately draw the game because Black would simply take the pawn. So, for our purposes, White will move his king to the right.
1.Kd3 Kc6. If you are forced to step away from a pawn in a position like this, it is a good general practice to always step straight back. Black cannot go to the side with 1.Kd3 Kb4?(bad move) 2.Kd4 and now Black can't get back in front of White's pawn, and he will queen. Back to the real analysis. 2.Kd4 Kd6. This move by Black keeps his King in front, and within reach, of White's pawn. It also prevents White from stepping his own King in front of his pawn. 3.c5+ Kc6. Black immediately steps back in front of White's pawn, any other move could potentially become disastrous. This move also forces White's next move. 4.Kc4(the only move that saves the pawn) Kc7. Again Black is applying the fundamentals, simply stepping back from the pawn. 5.Kd5 Kd7. Black remains adamant about keeping White's King from getting ahead of his pawn. 6.c6+ Kc7. Again forcing the White King to return behind the pawn if he want's to keep it. 7.Kc5 Kc8. By now you should see a trend. KEEP STEPPING BACKWARDS. 8.Kd6 Kd8. Still blocking out the King. 9.c7+ Kc8 Now the game is completely a draw. If White's King moves away from the pawn, Black simply takes it, and the game is drawn. What happens when White continues to defend the pawn? 9.Kc6. The game is drawn! Black is not in checkmate, but has no legal moves, therefore, he is in Stalemate, and the game is a tie.
Black to move in the original position is a stalemate for exactly the same reason. 1...Kd6. Stepping backwards for the draw! 2.Kd4 Kd6. Another take-home point. When White's King makes an attempt to get around his pawn, Black NEEDS to step directly in front of him. There is a tricky concept called Opposition, where you are trying to keep the other King from advancing forward, and for this drawing technique to work, Black needs to keep it, meaning he needs to stunt the White King from moving forward as much as possible. The rest of this is just a copy-cat version of the White-to-Move version. 3.c5+ Kc6 4.Kc4 Kc7 5.Kd5 Kd7 6.c6+ Kc7 7.Kc5 Kc8 8.Kd6 Kd8 9.c7+ Kc8 10.Kc6 Stalemate. This is the sort of knowledge that will frustrate the hell out of somebody thinking they have a won game.