A map of algebras inducing a bijection on maximal ideals is a bijection

Question

Choose an algebraically closed coefficient field $k$. Denote by $A$ a finitely generated integral domain over $k$. Let $f$ be a homomorphism of $k$-algebras $A\rightarrow A$ such that the pullback map from set of maximal ideals of $A$ to itself is a bijection (for well-definedness see https://math.stackexchange.com/a/107886/693243). Is $f$ itself a bijection?

EDIT: sorry, I meant $k$ to have characteristic $0$.

Answer 1

No. For instance, if $k$ has characteristic $p>0$ and $A=k[x]$, then the map $f$ given by $x\mapsto x^p$ is bijective on maximal ideals (since the $p$th power map on $k$ is a bijection) but $f$ is not surjective.