Let's write an analytic surjection $f\colon D^n\longrightarrow D^n$ that fixes the origin $f(0)=0$, where $D^n=\{x\in\mathbb{R}^n\colon|x|\leq1\}$. Is it possible that $f$ is not an open map?
Rephrased, is an analytic surjection from $D^n$ to itself with $f(0)=0$ necessarily an open map?