  This paper deals with the extension of a classical theorem by R. Phelps on
the G\^ateaux differentiability of Lipschitz functions on separable Banach
spaces to the non-separable case. The extension of the theorem is not possible
for general non-separable Banach spaces, as shown by Larman. Therefore, we will
work on some important non-separable spaces. We consider a norm of the
non-separable space $\ell^{\infty}(\mathbb{R})$, showing that it complies with
the aforementioned theorem thesis. We also consider other examples as
$L^{\infty}(\mathbb{R})$ and $NBV[a,b]$, showing in all cases their sets of
$G$-differentiability and some properties of these sets. All of the above,
closely following the assumptions in Phelps Theorem, which will extend in the
case separable to functions with rank over the Asplund spaces and in the
non-separable case for projective limits and locally-Lipschitz cylindrical
functions.
