  In this note, we will consider the question of local equivalence of analytic
functions which fix the origin and are tangent to the identity, as well as the
question of flows of analytic vector fields. All mappings and equivalences are
considered in the non-archimedean context e.g. all norms can be considered
$p$-adic norms. We show that any two mappings $f$ and $g$ which are formally
equivalent are also analytically equivalent, and we show that analytic vector
fields generate analytic flows. We consider the related questions of roots and
centralizers for analytic mappings. In this setting, anything which can be done
formally can also be done analytically.
