The Tannaka-Krein duality gives us that a necessary and sufficient and necessary condition for a $\mathbb{C}$-linear monoidal category to be equivalent to the category of continous, finite dimensional complex representations of of a compact group and even provides us with a recipe for constructing said group. The condition on the category is that the category ought to be semisimple and satisfy Schur's Lemma.
My question is how does the situation look like if we replace $\mathbb{C}$ with say $\mathbb{C}_p$ (or $\mathbb{Q}_p$). The first problem is that $p$-adic representations of compact groups need not be semisimple. If it helps, I'm quite happy to restrict myself to profinite groups.
Well if you are interested in characterizing categories that are representations of compact groups, then the answer can depend a lot on the relationship between the group and the field, and their topologies.
For example if we take $G = \mathbb{Z}_p$ then over $\mathbb{C}$ or $\mathbb{Q}_\ell$ for $\ell \ne p$ every representation of $G$ must be locally constant, and therefore everything is semisimple. But over $\mathbb{Q}_p$ we have non-semisimple representations like $a \to \left[ {\begin{array}{cc} 1 & a \\ 0 & 1 \\ \end{array} } \right]$. And in general the behavior for a fixed group can vary more wildly over different topological fields.
Going the other direction -- starting with a nice category of representations and trying to characterize the group. There is an algebrogeometric version of Tannaka-Krein theory due to Saavedra-Rivano, and later refined by Deligne and Milne, which works pretty well over any field. Here though semisimplicity in characteristic zero corresponds to a reductiveness condition on the group, rather than compactness.