We introduce a rigid analytification of the quantized coordinate algebra of a semisimple algebraic group and a quantized Arens-Michael envelope of the enveloping algebra of the corresponding Lie algebra, working over a non-archimedean field and when q is not a root of unity. We show that these analytic quantum groups are topological Hopf algebras and Fréchet-Stein algebras. We then introduce an analogue of the BGG category $O$ for the quantum Arens-Michael envelope and show that it is equivalent to the category $O$ for the corresponding quantized enveloping algebra.