Let X be an infinite-dimensional Banach space, and let A be a Cp Lipschitz bounded starlike body (for instance the unit ball of a smooth norm). We prove that:
(1) the boundary ∂A is Cp Lipschitz contractible;
(2) there is a Cp Lipschitz retraction from A onto ∂A;
(3) there is a Cp Lipschitz map T : A → A with no approximate fixed points.