Let $H$ be a connected, unipotent linear algebraic group defined over a local field $k$. Let $H \times_k X \rightarrow X$ be an action of $H$ on an irreducible, affine $k$-variety $X$ which is defined over $k$. Then the orbits of the group action $H(\overline{k}) \times X(\overline{k}) \rightarrow X(\overline{k})$ are Zariski-closed in $X(\overline{k})$.

What about the orbits of the group action $H(k) \times X(k) \rightarrow X(k)$? By the general theory of analytic manifolds, the orbits are locally closed subvarieties of $X(k)$. Are these orbits always closed, as in the algebraically closed case?