We prove that every Banach space containing an isomorphic copy of
c0 fails to have the fixed-point property for asymptotically
nonexpansive mappings with respect to some locally convex topology
which is coarser than the weak topology. If the copy of c0 is asymptotically isometric, this result can be improved, because
we can prove the failure of the fixed-point property for
nonexpansive mappings.