  In this paper we present a new construction of analytic analogues of quantum
groups over non-Archimedean fields and construct braided monoidal categories of
their representations. We do this by constructing analytic Nichols algebras and
use Majid's double-bosonisation construction to glue them together. We then go
on to study the rigidity of these analytic quantum groups as algebra
deformations of completed enveloping algebras through bounded cohomology. This
provides the first steps towards a $p$-adic Drinfel'd-Kohno Theorem, which
should relate this work to Furusho's $p$-adic Drinfel'd associators. Finally,
we adapt these constructions to working over Archimedean fields.
