For a profinite group G and a rigid analytic space X, we study when an OX (X)linear representation V of G admits a lattice, i.e. an OX (X )-linear model for a suitable formal model X of X in the sense of Berthelot. We give a positive answer, under mild assumptions, when X is a "wide open" space. As a consequence, we are able to describe explicit open rational subdomains of X over which V is constant after reduction modulo a power of p. We give applications in two different directions. First, we prove explicit results on the reduction modulo powers of p of sheaves of crystalline and semistable representations of fixed weight. Second, we focus on the sheaves of Galois representations on eigenvarieties, which are important examples of wide open spaces thanks to a result of Bellaïche and Chenevier. We give an application of our main results to the pseudorepresentation carried by the Coleman-Mazur eigencurve, which can be made explicit whenever equations for a rational subdomain of the eigencurve are given.