  We define "t-stratifications", a strong notion of stratifications for
Henselian valued fields $K$ of equi-characteristic 0, and prove that they
exist. In contrast to classical stratifications in Archimedean fields,
t-stratifications also contain non-local information about the stratified sets.
For example, they do not only see the singularities in the valued field, but
also those in the residue field.
  Like Whitney stratifications, t-stratifications exist for different classes
of subsets of $K^n$, e.g. algebraic subvarieties or certain classes of analytic
subsets. The general framework are definable sets (in the sense of model
theory) in a language that satisfies certain hypotheses.
  We give two applications. First, we show that t-stratifications in suitable
valued fields $K$ induce classical Whitney stratifications in $\Bbb R$ or $\Bbb
C$; in particular, the existence of t-stratifications implies the existence of
Whitney stratifications. This uses methods of non-standard analysis.
  Second, we show how, using t-stratifications, one can determine the
ultra-metric isometry type of definable subsets of $\Bbb Z_p^n$ for $p$
sufficiently big. For those $p$, this proves a conjecture stated in a previous
article. In particular, this yields a new, geometric proof of the rationality
of Poincar\'e series.
