We characterize amenability of subspaces of C(S), where S is a semigroup, in terms of fixed point properties of nonexpansive (1-Lipschitz) actions. In particular, using the notion of fragmentability, we give a complete characterization of semitopological semigroups with a left invariant mean on the space WAP(S) of weakly almost periodic functions on S that answers a question of [A.T.-M. Lau, Y. Zhang, J. Funct. Anal. 263 (2012), 2949–2977] in the affirmative. We also use Bruck’s method to show the existence of nonexpansive retractions onto common fixed point sets of S-actions and apply a fixed point theorem for semigroups with a LIM on WAP(S) to obtain nonlinear ergodic theorems in the spirit of [A.T.-M. Lau, N. Shioji, W. Takahashi, J. Funct. Anal. 161 (1999), 62–75].