In this talk I will discuss non-archimedean orbifolds covered by Mumford curves; our viewpoint to approach them is through the theory of p-adic orbifolds due to Yves André. After briefly explaining general ideas I will give three applications, first two of which are by joint-work with Gunther Cornelissen and Aristeides Kontogeorgis: (1) (char=p) sharp bound for the order of automorphism groups of Mumford curves in positive characteristic, (2) (char=p) determination of automorphism group of Drinfeld modular curves, and (3) (char=0) classification of p-adic schwarzian triangle groups related to Schottky-Mumford uniformization.