If $\Omega$ is a cube in $\mathbb{R}^n$ and $f\in C^1(\overline\Omega)$. By reflection one can extend such a function to all of $\mathbb{R}^n$ and the extenstion is in $C^1(\mathbb{R}^n)$. If $\Omega$ is a polygon, has piecewise $C^1$ boundary (so edges and corneres are not to wild) or is a convex set this still seems to be possible. Can this be extended to arbitrary Lipschitz domains?
Are there examples and or references for these cases (starting from polygons)?
As George Lowther pointed out, such an extension is possible for any quasiconvex domains (in particular, for any Lipschitz domain). This is the main result in a short paper by Whitney from 1934:
Property P is what we now call quasiconvexity:
This result, and many later developments, are presented in section 2.5 of the book