Let $G$ be a connected algebraic group over an algebraically closed field $\overline{k}$ acting on an irreducible variety $X$. A geometric quotient is a morphism of varieties $\pi: X \rightarrow X/\sim$ which on closed points (that is, as a morphism of classical varieties) satisfy the following:

(i): $\pi$ is a surjective open map, and the fibres are exactly the $G$-orbits of $X$.

(ii): for any open set $U$ of $X/\sim$, the ring homomorphism $\pi^{\ast}: \overline{k}[U] \rightarrow \overline{k}[\pi^{-1}U]$ is an isomorphism onto the $G$-fixed points of codomain.

Rosenlicht's theorem says that there exists a $G$-stable open subset of $X$ for which the geometric quotient exists.

Is there any generalization of Rosenlicht's theorem for when $G$ and $X$ are defined over an arbitrary field? The case I'm interested in is when $G$ and $X$ are geometrically connected subgroups of upper triangular unipotent matrices over a $p$-adic field $F$ (so all orbits are closed), and $X$ is normalized by $G$.

perfectfield $k$. You can choose a $G$-stable open subset $U$ of $X$ (for which the geometric quotient exists) defined over $k$. It seems that your geometric quotient $U/\sim$ is has no automorphisms (compatible with the identity automorphism of $U$). If $U/\sim$ is quasi-projective, you can construct a $k$-model of it by Galois descent. (I hope all this is correct....) $\endgroup$