  We give conclusive answers to some questions about definability in analytic
languages that arose shortly after the work by Denef and van den Dries, [DD],
on $p$-adic subanalytic sets, and we continue the study of non-archimedean
fields with analytic structure of [LR3], [CLR1] and [CL1]. We show that the
language $L_K$ consisting of the language of valued fields together with all
strictly convergent power series over a complete, rank one valued field $K$ can
be expanded, in a definitial way, to a larger language corresponding to an
analytic structure (with separated power series) from [CL1], hence inheriting
all properties from loc. cit., including geometric properties for the definable
sets like certain forms of quantifier elimination. Our expansion comes from
adding specific, existentially definable functions, which are solutions of
certain henselian systems of equations. Moreover, we show that, even when $K$
is algebraically closed, one does not have quantifier elimination in $L_K$
itself, and hence, passing to expansions is unavoidable in general. We pursue
this study in the wider generality of extending non-separated power series
rings to separated ones, and give new examples, in particular of the analytic
structure over $\mathbb{Z}[[t]]$ that can be interpreted and understood now in
all complete valued fields. In a separate direction, we show in rather large
generality that Weierstrass preparation implies Weierstrass division.
