The Bott periodicity theorem can be formulated as the existence of homotopy equivalences $\Omega^2(KU)\equiv KU$ and $\Omega^8(KO)=KO$. I always wondered whether this theorem could also be transferred to the other completions of $Q$, i. e. the $p$-adic fields. Of course, since the topology of $\mathbb Q^p$ behaves rather incompatibly with the topology of, say, $CW$ complexes, it's not really clear how to make sense of that. (The classifying space of the topological group should have the same homotopy groups as the classifying space constructed from the underlying discrete group, if I'm not mistaken). Therefore a more sophisticated approach (possibly via $p$-adic rigid geometry) might be necessary. So, does there exist anything which could be considered as the $p$-adic analogue of Bott periodicity?

As I understood it, the underlying algebraic reason for the lengths of the period in the real and complex case, respectively) is that the sequence of Clifford algebras $Cl(k,n)$ exhibits the same periodicity behaviour up to Morita equivalence (precisely, $Cl(k+8,\mathbb R )=M(Cl(k,\mathbb R),8)$ and $Cl(k+2,\mathbb R )=M(Cl(k,\mathbb R),2)$. A quick check in the literature shows that the clifford algebras over $\mathbb Q_p$ are also periodic of length $2$ if $p\equiv 3 \mod 4$, $4$ if $p \equiv 1 \mod 4$ and $8$ for $p=2$, so at least at this algebraic level, Bott periodicity is present.