I understand that there is a definition of p-adic Banach algebras and that a significant amount of functional analysis can be developed in the non-archimedean setting. Is there a p-adic version of C*-algebras? If so, is there an analogue of the GNS construction?
There exists a complete theory of non-Archimedean commutative Banach algebras. In particular, there are conditions under which an algebra is isomorphic to the algebra of continuous functions. For the commutative case, they can be seen as the counterparts of the $C^*$ condition. Note that there is no natural involution in the p-adic case. See
V. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, AMS, 1990 (the above conditions are in Corollary 9.2.7);
A. Escassut, Ultrametric Banach algebras, World Scientific, 2003.
For the noncommutative case, very little is known.