A remark on smooth images of Banach spaces

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∞-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]).