Smooth Lipschitz retractions of starlike bodies onto their boundaries in infinite-dimensional Banach spaces

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 partial derivativeA is C-P Lipschitz contractible; (2) there is a C-P Lipschitz retraction from A onto partial derivativeA; (3) there is a C-P Lipschitz map T : A --> A with no approximate fixed points.