On extremal nonexpansive mappings

We study the extremality of nonexpansive mappings on a nonempty bounded closed and convex subset of a normed space (therein specific Banach spaces). We show that surjective isometries are extremal in this sense for many Banach spaces, including Banach spaces with the Radon-Nikodym property and all $C(K)$-spaces for compact Hausdorff $K.$ We also conclude that the typical, in the sense of Baire category, nonexpansive mapping is close to being extremal.