Random generation in semisimple algebraic groups over local fields

Let G be a semisimple algebraic group over a local field K of characteristic p. If the universal covering map Gsc→G is inseparable then no open subgroup of G(K) is finitely generated. Otherwise, for any compact open subgroup with probability 1 two randomly chosen elements generate an open subgroup of G(K). Our main tool is a recent theorem by R. Pink characterizing compact groups linear over a local field.