  The purpose of this paper is to introduce new definitions of H\"ormander
classes for pseudo-differential operators over the compact group of $p$-adic
integers. Our definitions possess a symbolic calculus, asymptotic expansions
and parametrices, together with an interesting relation with the infinite
matrices algebras studied by K. Gr\"ochenig and S. Jaffard. Also, we show how
our definition of H\"ormander classes is related to the definition given in the
toroidal case by M. Ruzhansky and V. Turunen. In order to show the special
properties of our definition, in a later work, we will study several spectral
properties in terms of the symbol for pseudo-differential operators in the
H\"ormander classes here defined.
