Is there an example for a Lipschitz map $f:\mathbb{R}^n\to\mathbb{R}^m$ which is differentiable at $x_o$, with $\operatorname{rank} Df(x_o)=k$, such that there is no open neighbourhood $U$ of $x_0$ where $\operatorname{rank} Df \ge k$ (wherever it is defined on $U$).
Note that by the Rademacher theorem $f$ is differentiable a.e. Of course, if $f$ was $C^1$, then such a neighbourhood would always exist.
I am not even sure what happens when $m=n=1$.
Let $f\colon\Bbb R\to\Bbb R$ be defined as $f(x)=x+x^2\sin(x^{-1})$ if $x\ne0$, $f(0)=0$. $f$ is differentiable, but not $C^1$, and globally Lipschitz. $f'(0)=1$, but on every neighborhood of $0$ there are points where $f'$ vanishes.