Abstract
Let X be a non-separable super-reflexive Banach space. Then for any separable Banach space Y of dimension at least two there exists a C-infinity-smooth surjective mapping f : X -> Y such that the restriction of f onto any separable subspace of X fails to be surjective.
This solves a problem posed by Aron, Jaramillo, and Ransford (Problem 186 in the book [5]).