The Yokonuma-Hecke algebras are quotients of the modular framed braid group and they support Markov traces. In this paper, which is sequel to Juyumaya and Lambropoulou (2007) [6], we explore further the structures of the p-adic framed braids and the p-adic Yokonuma-Hecke algebras constructed by Juyumaya and Lambropoulou (2007) [6], by means of dense sub-structures approximating p-adic elements. We also construct a p-adic Markov trace on the p-adic Yokonuma-Hecke algebras and approximate the values of the p-adic trace on p-adic elements. Surprisingly, the Markov traces do not re-scale directly to yield isotopy invariants of framed links. This leads to imposing the 'E-condition' on the trace parameters. For solutions of the 'E-system' we then define 2-variable isotopy invariants of modular framed links. These lift to p-adic isotopy invariants of classical framed links. The Yokonuma-Hecke algebras have topological interpretations in the context of framed knots, of classical knots of singular knots and of transverse knots. (C) 2012 Elsevier Inc. All rights reserved.