I think such examples exist, take as $f:Y\rightarrow \mathbb P^1$ any family of supersingular abelian varieties over $\overline{\mathbb F_p}$ which is non-constant but constant up-to isogeny (e.g. take the Moret-Bailly pencil of supersingular abelian surfaces).
You cannot have a non-constant morphism $\mathbb P^1\rightarrow Y$, since the group $Y(\overline{\mathbb F_p}(t))$ is torsion (since the family is isotrivial up to isogeny) and any non-constant morphism $\mathbb P^1\rightarrow Y$ would correspond to a non-torsion element in $Y(\overline{\mathbb F_p}(t))$.