Do smooth projective maps to $\mathbb{P}^1$ in positive characteristic have sections?

Question

In response to the affirmative answer to this question using symplectic methods, I am wondering if the equivalent statement holds in positive characteristic?

Explicitly, over an algebraically closed field $k$ of positive characteristic $p > 0$, does every smooth projective morphism $f : X \to \mathbb{P}^1$ admit a section?

If we do not require $k$ to be separably closed, the question becomes uninteresting because of examples such as $\mathbb{P}^1_{k'} \to \mathbb{P}^1_k$ for any finite separable $k'/k$.

My guess is there should exist an example with no section. My idea was to find a surface admitting a nontrivial elliptic fibration over $\mathbb{P}^1$ with smooth fibers. Do there exist such counterexamples?

Answer 1

EDIT: It was wrong.

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 $Y(\overline{\mathbb F_p}(t))$.