This coq library aims to formalize a substantial body of mathematics using the univalent point of view.
Formalization of p-adic numbers, described in
Álvaro Pelayo, Vladimir Voevodsky, and Michael A. Warren A univalent formalization of the p-adic numbers Math. Struct. in Comp. Science (2015), vol. 25, pp. 1147–1171. doi:10.1017/S0960129514000541
Also see Álvaro Pelayo, Vladimir Voevodsky and Michael A. Warren, A preliminary univalent formalization of the p-adic numbers at arXiv.