Does Lipschitz imply locally surjective?

Question

Let $f:\mathbb C\to\mathbb C$ a non constant Lipschitzian function with parameter $k<1$, that is: for all $(x,y)\in\mathbb C^2$ one has $|f(x)-f(y)|<k|x-y|$. Is $f$ locally surjective that is: for every $a\in f(\mathbb C)$, there exists open sets $U,V$ such that $a\in U$ and $f(a)\in V$ with $f(U)=V$?