Let $k$ be a non-Archimedean local field, and let $G$ be an affine algebraic group over $k$. The literature suggests that the following are well established facts:

The group $G(k)$ of $k$-rational points in $G$ inherits the structure of a locally profinite group w.r.t. the subspace topology, and in the case that $char(k)=0$ the group $G(k)$ also has the structure of an analytic manifold, turning it into a $p$-adic Lie group.

Unfortunately, I am not well versed in the theory of analytic manifolds, and its apparent interplay with algebraic geometry in the above setting. Taking this into account, I would like to ask clarifications and/or references for the above mentioned facts (which to me are folklore for now) and the following questions:

Keep the notation introduced so far. In what follows, $\overline{k}$ denotes the algebraic closure of $k$. Furthermore assume that $char(k)=0$.

1) What is the relation between the subspace topology on $G(k)$ inherited from $G(\overline{k})$ and the topology of its underlying analytic manifold?

2) More generally, if $V$ is an affine variety over $k$, does $V(k)$ possess the structure of an analytic manifold in any canonical way? If so, what is the relation between its underlying topology and the subspace topology inherited from $V(\overline{k})$?

3) Suppose $G$ acts $k$-morphically on an affine variety $V$ over $k$. The action morphism $G \times V \rightarrow V$ gives rise to an action map $G(k) \times V(k) \rightarrow V(k)$. Is this map continuous if both $G(k)$ and $V(k)$ are given the subspace topology inherited from $G(\overline{k})$ and $V(\overline{k})$ respectively, and $G(k) \times V(k)$ is given the product topology? In case question "2)" has a positive answer, what can be said about the same action map if $G(k)$ and $V(k)$ are both given the analytic manifold topology and $G(k) \times V(k)$ is given the product topology?

Thank you in advance.

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    $\begingroup$ (1) it's hard to answer if you don't say which topology you consider on $G(\bar{k})$. (2) yes if $V$ is smooth for the 1st question. (And yes to the 2nd if you endow $\hat{k}$ with the norm extending a norm on $k$ and topologize $V(\hat{k})$ accordingly) (3) again it's hard to answer but if varieties are endowed with the norm topology then regular maps are continuous, and this applies to the map $G(k)\times V(k)\to V(k)$; if $V$ is smooth this is also a $k$-analytic map. $\endgroup$ – YCor Nov 1 '19 at 0:56
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    $\begingroup$ I think that most of your questions are answered in Bjorn Poonen's book Rational points on varieties. I would look there. $\endgroup$ – Alex Youcis Nov 1 '19 at 9:49

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