Is there a $p$-adic version of Liouville theorem? That is, if a function $f$ is analytic and bounded in all $K$, a $p$-adic field (or more generally a complete non-archimedean field), has to be constant?
And does the theorem work for functions on $K^n$, or in $\mathbb{C}^n$?

The answer to the first question is yes, Liouville's theorem still holds for valued fields that are algebraically closed (this last part added after Pete Clark's comment below). See, for example, these lecture notes by William Cherry (in particular, see page 16).
