Descent for admissible representations of algebraic groups over local fields

Question

Let $G$ be a reductive group and $F$ be a local field which is a finite extension of $\mathbb{Q}_p$. Assume $\Pi$ is an irreducible smooth admissible representation of $G(F)$ over $\bar{\mathbb{Q}}_l$, where $l$ could be $p$. Can $\Pi$ always descend to a representation of $G(F)$ over a local field contained in $\bar{\mathbb{Q}}_l$? The motivation of the question is a similar question in Galois side: Does the image of a p-adic Galois representation always lie in a finite extension? and the $p$-adic Langlands program where we consider admissible unitary representations on a Banach space over a local field over $\mathbb{Q}_p$.