Non-uniformly continuous nearest point maps

We construct a Banach space satisfying that the nearest point map (also called proximity mapping or metric projection) onto any non-singleton compact and convex subset is continuous but not uniformly continuous. The space we construct is locally uniformly convex, which ensures the continuity of all these nearest point maps. Moreover, we prove that every infinite-dimensional separable Banach space is arbitrarily close (in the Banach-Mazur distance) to one satisfying the above conditions.