I know the following classical fact about rational points over a perfect, infinite field (p.s. everything is about linear algbaic groups, i.e. all groups are affine):
If $G$ is a connected group defined over $F$, where $F$ is an infinite $\textbf{perfect}$ field and $k$ its algebraic closure, then $G(F)$ is Zariski dense in $G(k)$. This is a classical result due to Rosenlicht. If $G$ is reductive, then the result is also valid for non-perfect infinite field, which is a famous result due to Grothendieck.
But while working over general local fields including characteristic $p$ fields, sometimes rationality properties of unipotent subgroups would be very helpful. So far I know nothing related to this question.
First, if $G$ is an arbitrary connected $F$-group, when $F$ is non-perfect, its unipotent radical may not be defined over $F$. So I want to ask if there is any result related to characteristic $p$ local fields for this situation?
Secondly, if $U$ is a nontrivial unipotent $F$-group with $F$ a characteristic $p$ local field, what general property can we say about its $F$-points? Is $U(F)$ always nontrivial? or any further density results?
Thank a lot in advance for your help! And forgive me that my questions may seem to be very stupid for experts, since I'm a beginner in the study of algebraic groups! :)