Smooth extension of functions on a certain class of non-separable Banach spaces

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 $\text{Lip}(g)\le C\text{Lip}(f)$ (with $C$ depending only on the space $X$). This is the case for a Banach space $X$ bi-Lipschitz homeomorphic to a subset of $c_0(\mathrm{\Gamma })$, for some set $\mathrm{\Gamma }$, 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 $f:Y\to \text{Real}$, 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 $X$.