Let $k$ be a finite extension of the p-adic number field $Q_p$ and G be a connected algebraic (not affine) group over $k$. It is well-known (see e.g. [1] Proposition 3.1) that G decomposes as $1\rightarrow G_{aff}\cap G_{ant}\rightarrow G_{aff}\times G_{ant}\rightarrow G \rightarrow 1,$ where $G_{aff}$ is the smallest normal connected affine subgroup of $G$ such that $G/G_{aff}$ is an abelian variety and $G_{ant}$ is the smallest normal subgroup such that $G/G_{ant}$ is affine.

Then $(G_{ant})_{aff}$ is a connected commutative affine algebraic group, and hence admits a unique decomposition $S\times U$, where $S$ is a torus and $U$ is connected and unipotent. Let $G'$ be a quasi-complement of $S$ in $G_{aff}/R_u$, where $R_u$ is the unipotent radical of $G_{aff}$. If $\pi:G_{aff}\rightarrow G_{aff}/R_u$ is the quotient map, then $\tilde{G}=\pi^{-1}(G')$ is a quasi-complement of $S$ in $G_{aff}$. i.e. $G_{aff}=S\tilde{G}$ with $S \cap\tilde{G}$ is finite. Together with above exact sequence we get an isogeny (see [2] Proposition 2.2)

$\phi:(\tilde{G}\times G_{ant})/U \rightarrow G$ such that the kernel of $\phi$ is $(\tilde{G}\cap G_{ant})/U$. Note that $U$ is the connected component of $\tilde{G}\cap G_{ant}$.

Q1: Is $\phi((\tilde{G}\times G_{ant})/U)(k))=\frac{(\tilde{G}\times G_{ant})/U)(k)}{(\tilde{G}\cap G_{ant})/U)(k)}$ closed in $G(k)$ w.r.t. analytic topology induced from $k$.

In general $\phi$ is not surjective on $k$-rational points. We have the following exact sequence $1 \rightarrow ((\tilde{G}\cap G_{ant})/U)(k)\rightarrow (\tilde{G}\times G_{ant})/U)(k) \rightarrow G(k)\rightarrow H^1(k_s/k, (\tilde{G}\cap G_{ant})/U)$, where $H^1$ is the first Galois cohomology. Since $(\tilde{G}\cap G_{ant})/U$ is a central finite affine algebraic group, $H^1$ is finite abelien group.

Q2: Does there exist a finite Galois extension $K$ of $k$ or $Q_p$ such that $\phi$ is indeed surjective on $K$-rational points? I think this is weaker than $H^1(k_s/k, (\tilde{G}\cap G_{ant})/U)=0$. If it is necessary, we may change the quasi-complement $\tilde{G}$.

[1] Anti-affine algebraic groups, Michel Brion, Journal of Algebra

[2] Principal bundles, quasi-abelian varieties and structure of algebraic groups, Carlos Sancho de Salasa, , Fernando Sancho de Salas, Journal of Algebra

I am very sorry for heavy notation.