Abstract. If is a a scheme of finite type over a local field F, and is a proper smooth family, then to each rational point one can assign an extension of the absolute Galois group of F by the geometric fundamental group G of the fibre . If F has uniformiser , and residue characteristic p, we show that the corresponding extension of the absolute Galois group of by the maximal prime to p quotient of G is locally constant in the -adic topology of . We give a similar result in the case of non-proper families, and families over -adic analytic spaces.