  We study free topological groups defined over uniform spaces in some
subclasses of the class NA of non-archimedean groups. Our descriptions of the
corresponding topologies show that for metrizable uniformities the
corresponding free balanced, free abelian and free Boolean NA groups are also
metrizable. Graev type ultra-metrics determine the corresponding free
topologies. Such results are in a striking contrast with free balanced and free
abelian topological groups cases (in standard varieties).
  Another contrasting advantage is that the induced topological group actions
on free abelian NA groups frequently remain continuous. One of the main
applications is: any epimorphism in the category NA must be dense. Moreover,
the same methods improve the following result of T.H. Fay : the inclusion of a
proper open subgroup H into G is not an epimorphism in the category of all
Hausdorff topological groups. A key tool in the proofs is Pestov's test of
epimorphisms.
  Our results provide a convenient way to produce surjectively universal NA
abelian and balanced groups. In particular, we unify and strengthen some recent
results of Gao and Gao-Xuan as well as classical results about profinite groups
which go back to Iwasawa and Gildenhuys-Lim.
