Strengthening Ax-Grothendieck

The question was cross-posted from Math.SE: https://math.stackexchange.com/questions/4566017/strengthening-ax-grothendieck

The question is simple. The Ax-Grothendieck theorem says a polynomial map $p\colon\mathbb C^n\to\mathbb C^n$ that is injective is also surjective.

Is assuming $p$ has finite fibers enough?

I can't come up with any easy counter-examples, but the proof I know using finite fields does not work.