Let $f=\sum_{n,m \geq 0}^{\infty}[a_{nm}]p^ny^m \in \mathbb Z_p[[y]]$, where $a_{nm} \in \mathbb F_p$ and $[\cdot]$ means the Teichmüller lifting. Define $I(f)=\sum_{n \geq 0}[a_{nn}]p^nt^n \in \mathbb Z_p [[t]]$.

If $f$ is a rational function of $y$, must $I(f)$ lie in a finite extension of $\mathbb Q_p(t)$?

Motivation: the equal characteristic case is true (and the generalization to several variable case is also true in positive characteristic), for example see "Intégration sur un cycle évanescent" by Deligne, which gives a beautiful geometric proof.