  Let $K$ be a complete non-Archimedean field $K$ with separated power series,
treated in the analytic Denef--Pas language. We prove the existence of
definable retractions onto an arbitrary closed definable subset of $K^{n}$,
whereby definable non-Archimedean versions of the extension theorems by
Tietze--Urysohn and Dugundji follow directly. We reduce the problem to the case
of a simple normal crossing divisor, relying on our closedness theorem and
desingularization of terms. The latter result is established by means of the
following tools: elimination of valued field quantifiers (due to
Cluckers--Lipshitz--Robinson), embedded resolution of singularities by blowing
up (due to Bierstone--Milman or Temkin), the technique of quasi-rational
subdomains (due to Lipshitz--Robinson) and our closedness theorem.
