Following constructions in rigid analytic geometry, we introduce a theory of $p$-adic analytic quantum groups. We first define Fréchet completions $\backslash wideparenUq(g)$ and $\backslash wideparenOq(G)$ of the quantized enveloping algebra of a semisimple Lie algebra $g$ and the quantized coordinate ring of the corresponding semisimple algebraic group $G$ respectively. We consider these to be quantum analogues of the Arens-Michael envelope of the enveloping algebra $U(g)$ and of the algebra of rigid analytic functions on the rigid analytification of $G$ respectively. We show that these algebras are topological Hopf algebras and, by adapting techniques extracted from work of Ardakov-Wadsley, Schmidt and Emerton in $p$-adic representation theory, we also show that they are Fréchet-Stein algebras and use this to investigate an analogue of category $O$ for $\backslash wideparenUq(g)$.

We then introduce a $p$-adic analytic analogue of Backelin and Kremnizer's construction of the quantum flag variety of a semisimple algebraic group, using a Banach completion of $\backslash wideparenOq(G)$. Our main result is a Beilinson-Bernstein localisation theorem in this context. We define a category of $\lambda$-twisted $D$-modules on this analytic quantum flag variety. This category has a distinguished object $\backslash Dq\lambda ^$ which plays the role of the sheaf of $\lambda$-twisted differential operators. We show that when $\lambda$ is regular and dominant, the global section functor gives an equivalence of categories between the coherent $\lambda$-twisted $D$-modules and the finitely presented modules over the global sections of $Dq\lambda ^$.

The construction of this analytic quantum flag variety involves working with Banach comodules over the Banach completion $\backslash OqBhat$ of the quantum coordinate algebra of the Borel. Along the way, we also show that Banach comodules over $\backslash OqBhat$ can be naturally identified with what we call topologically integrable modules over the Banach completion of Lusztig's integral form of the quantum Borel.