Could the black king move to C5?
No, it is checkmate. Even if Pb4 is pinned, it still controls square c5, preventing the black king to move there.
Check rules 3.9.1 and 3.9.2 of the laws of chess.
3.9.1 The king is said to be 'in check' if it is attacked by one or more of the opponent's pieces, even if such pieces are constrained from moving to the square occupied by the king because they would then leave or place their own king in check.
No piece can be moved that will either expose the king of the same colour to check or leave that king in check.
1...Kc5 is illegal.