It is shown that (1) every infinite-dimensional Banach space admits a
C-1 Lipschitz map onto any separable Banach space, and (2) if the dual
of a separable Banach space X contains a normalized, weakly null Bana
ch-Saks sequence, then X admits a C-infinity map onto any separable Ba
nach space. Subsequently, we generalize these results to mappings onto
larger target spaces.