Assume $f\colon \mathbb{D}\to\mathbb{R}^2$ is a harmonic map and $x\notin f(\partial\mathbb{D})$. Is it true that $f^{-1}\{x\}$ is totally disconnected?

I hope that the answer is yes. But actually I need a "yes", with a CAT(0) space instead of $\mathbb{R}^2$.

So, I need a generalizable proof.

**P.S.** For CAT(0) target the answer is "no" --- if the target is a cone with large total angle then the tip of the cone might have a tree as an inverse image for harmonic map; this example is constructed in "Harmonic maps between flat surfaces with conical singularities" by Ernst Kuwert.