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. In fact, even a spectral theorem for a single operator is very recent for the p-adic case; see

A. N. Kochubei, Non-Archimedean normal operators, J. Math. Phys. 51, Article 023526 (2010).

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.