  Let us consider a Banach space $X$ with the property that every real-valued
Lipschitz function $f$ can be uniformly approximated by a Lipschitz,
$C^1$-smooth function $g$ with $\Lip(g)\le C \Lip(f)$ (with $C$ depending only
on the space $X$). This is the case for a Banach space $X$ bi-Lipschitz
homeomorphic to a subset of $c_0(\Gamma)$, for some set $\Gamma$, such that the
coordinate functions of the homeomorphism are $C^1$-smooth. Then, we prove that
for every closed subspace $Y\subset X$ and every $C^1$-smooth (Lipschitz)
function $f:Y\to\Real$, there is a $C^1$-smooth (Lipschitz, respectively)
extension of $f$ to $X$. We also study $C^1$-smooth extensions of real-valued
functions defined on closed subsets of $X$.
