It is not possible. The following result is true:

>**Theorem.** If $f:\mathbb{R}^n\to\mathbb{R}^n$ is differentiable and $|A|=0$, then $|f(A)|=0$.

Lipschitz mappings map sets of Lebesgue measure zero to the sets of Lebesgue measure zero. Therefore the above result is a consequence of the following:

>**Lemma.** If $f:\mathbb{R}^n\to\mathbb{R}^n$ is differentiable, then $\mathbb{R}^n=\bigcup_{i=1}^\infty E_i$ such that $f|_{E_i}$ is Lipschitz.

I will write a proof in a moment.