A Non-abelian Free Pro-pGroup Is Not Linear over a Local Field

In this paper we show that a (non-abelian) free pro-pgroup cannot be obtained as a closed subgroup ofGLn(F), whereFis a non-archimedean local field andnis arbitrary. Using a theorem of E. I. Zelʹmanov we deduce some group theoretic properties of linear pro-pgroups over a local field. Our main tool is a recent theorem by Pink characterizing compact subgroups ofGLn(F)