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 {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 Y is smooth (i.e. the Borel structure on G Y is countably separated), then topological amenability of G Y is equivalent to amenability of all point stabilizers in G.