In this paper, we investigate the common fixed point property for commutative nonexpansive mappings on $\tau$-compact convex sets in normed and Banach spaces, where $\tau$ is a Hausdorff topological vector space topology that is weaker than the norm topology. As a consequence of our main results, we obtain that the set of common fixed points of any commutative family of nonexpansive self-mappings of a nonempty clm-compact (resp. weak* compact) convex subset C of $L_1(\mu)$ with a $\sigma$-finite $\mu$ (resp. the James space $J_0$) is a nonempty nonexpansive retract of C.
