What is the topological dimension of a (locally analytic) $p$-adic manifold over a non Archimedean field $K$?
Is the topological dimension of $K^n$, $n$?
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$p$-adic numbers are locally compact, Hausdorff and totally disconnected (see this nLab page), hence they are zero-dimensional. This means that---at least naively---topological dimension of $p$-adic manifolds doesn't work as you'd expect from real or complex manifolds. However, there are ways to do analytic geometry over the $p$-adic numbers, see e.g. this stackexchange question and the references on this nLab page.