# 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 $\mathbb{C}^n$?

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"Analytic" means what? Since the space is zero-dimensional, there are certainly sets $U$ that are open and closed, not empty and not the whole space. The characteristic function of such a set is identically constant on a neighborhood of every point. Note that the answer says "entire" and not "analytic"... – GEdgar Jan 14 '12 at 14:46

Note that Cherry's result is for a valued field which is algebraically closed, e.g. $\mathbb{C}_p$. If I remember correctly, there are nonconstant bounded entire functions on $\mathbb{Q}_p$ (or any finite extension thereof), just as there are on $\mathbb{R}$. I believe Alain Robert's GTM has a discussion of this. – Pete L. Clark Jan 14 '12 at 5:01