Amenable actions of real and $p$-adic algebraic groups

Let $K$ be a locally compact field of characteristic 0. Let $G$ be a linear algebraic group defined over $K$, acting algebraically on an algebraic variety $V$. We prove that the action of $G(K)$ (the group of $K$-rational points of $G$) on $V(K)$ is topologically amenable, if and only if all points stabilizers in $G(K)$ are solvable-by-compact. This follows by combining a result by Borel-Serre \cite{BoSe} with the following fact: let $G$ be a second countable locally compact group acting continuously on a second countable locally compact space $Y$. If the action $G\curvearrowright Y$ is smooth (i.e. the Borel structure on $G\mathrm{\backslash }Y$ is countably separated), then topological amenability of $G\curvearrowright Y$ is equivalent to amenability of all point stabilizers in $G$.