Here is the proof of the inverse function theorem for differentiable, non $C^1$ mappings, supposing that $Df$ has maximum rank https://terrytao.wordpress.com/2011/09/12/the-inverse-function-theorem-for-everywhere-differentiable-maps/ .
The thing I can't prove, and that is left to the reader, is that $f^{-1}$ is differentiable. Infact by setting $f^{-1}(y+k)=x+h, f^{-1}(y)=x$, I get $f^{-1}(y+k)-f^{-1}(y)-Df(f^{-1}(y))^{-1}(k) = -Df(f^{-1}(y))^{-1}(f(x+h)-f(x)-Df(x)(h))=o(|h|)=o(|f^{-1}(y+k)-f^{-1}(y)|)$, and i can't prove without continuity of $Df$ at $x$ that $o(|f^{-1}(y+k)-f^{-1}(y)|)=o(|k|)$.
Any help or reference would be much appreciated, thanks in advance!