Let $M^n$ and $N^n$ be two compact complex (or complex projective) manifolds. Let $f: M\to N$ be a holomorphic map of degree one. How to prove that for each $x\in N$ the set $f^{-1}(x)$ is simply-connected?

Added. The above statement would follow from a different one:
There is an $\varepsilon$-neighbourhood $U_x$ of $x$ such that $f^{-1}(x)$ is the deformation retract of $f^{-1}(U_x)$. This second statement seems very plausible but I don't know how to prove it.

Indeed, since $U_x$ and $f^{-1}(U_x)$ are birational, they have same fundamental group (Griffiths Harris page 474), i.e $f^{-1}(U_x)$ is simply connected.