Does the inverse function theorem hold for everywhere differentiable maps?

The usual reference to the proof is A. V. Cernavskii in "Finite-to-one open mappings of manifolds", Mat. Sb. (N.S.), 65(107) (1964), 357–369 and "Addendum to the paper "Finite-to-one open mappings of manifolds"", Mat. Sb. (N.S.), 66(108) (1965), 471–472. If I remember it correctly, he does not state it explicitly, but it follows from what is there.

For another proof that your question has a positive answer, you may look at the folowing paper by Jean Saint Raymond: http://www.math.jussieu.fr/~raymond/preprints/inversion.dvi

It seems that he was not aware of the reference given by David Preiss above - his proof is in English so at least it should avoid you having to learn Russian just for that...

From http://arxiv.org/abs/1011.1288 by Ivar Ekeland:

I present an inverse function theorem for differentiable maps between Frechet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue's dominated convergence theorem and Ekeland's variational principle. As a consequence, the assumptions are substantially weakened: the map F to be inverted is not required to be C^2, or even C^1, or even Frechet-differentiable.