  We introduce a $p$-adic analytic analogue of Backelin and Kremnizer's
construction of the quantum flag variety of a semisimple algebraic group, when
$q$ is not a root of unity and $| q-1|<1$. We then define a category of
$\lambda$-twisted $D$-modules on this analytic quantum flag variety. We show
that when $\lambda$ is regular and dominant and when the characteristic of the
residue field does not divide the order of the Weyl group, the global section
functor gives an equivalence of categories between the coherent
$\lambda$-twisted $D$-modules and the category of finitely generated modules
over $\widehat{U_q^\lambda}$, where the latter is a completion of the ad-finite
part of the quantum group with central character corresponding to $\lambda$.
Along the way, we also show that Banach comodules over the Banach completion
$\widehat{\mathcal{O}_q(B)}$ of the quantum coordinate algebra of the Borel can
be naturally identified with certain topologically integrable modules.
