A well-known open question is whether every countable collection of Lipschitz functions on a Banach space $X$ with separable dual has a common point of Fréchet differentiability. We show that the answer is positive for some infinite-dimensional $X$. Previously, even for collections consisting of two functions this has been known for finite-dimensional $X$ only (although for one function the answer is known to be affirmative in full generality). Our aims are achieved by introducing a new class of null sets in Banach spaces (called $\Gamma$-null sets), whose definition involves both the notions of category and measure, and showing that the required differentiability holds almost everywhere with respect to it. We even obtain existence of Fréchet derivatives of Lipschitz functions between certain infinite-dimensional Banach spaces; no such results have been known previously.
Our main result states that a Lipschitz map between separable Banach spaces is Fréchet differentiable $\Gamma$-almost everywhere provided that it is regularly Gâteaux differentiable $\Gamma$-almost everywhere and the Gâteaux derivatives stay within a norm separable space of operators. It is easy to see that Lipschitz maps of $X$ to spaces with the Radon-Nikodým property are Gâteaux differentiable $\Gamma$-almost everywhere. Moreover, Gâteaux differentiability implies regular Gâteaux differentiability with exception of another kind of negligible sets, so-called $\sigma$-porous sets. The answer to the question is therefore positive in every space in which every $\sigma$-porous set is $\Gamma$-null. We show that this holds for $C(K)$ with $K$ countable compact, the Tsirelson space and for all subspaces of $c_0$, but that it fails for Hilbert spaces.