Let $k$ be a field, $X$ be a connected smooth projective $k$-scheme. Let $f:X\rightarrow X$ be a finite $k$-morphism that is surjective on the underlying topological spaces. Suppose $f$ has degree $n$.

Is it possible that there does not exist a coherent locally free sheaf $F$ on $X$ such that $H^0(X, F)=n\,\mathrm{rank}(F)$?

I think for curves, one can find line bundles of any $H^0$.