  In this article, we introduce a notion of non-degeneracy, with respect to
certain Newton polyhedra, for rational functions over non-Archimedean locals
fields of arbitrary characteristic. We study the local zeta functions attached
to non-degenerate rational functions, we show the existence of a meromorphic
continuation for these zeta functions, as rational functions of $q^{-s}$, and
give explicit formulas. In contrast with the classical local zeta functions,
the meromorphic continuation of zeta functions for rational functions have
poles with positive and negative real parts.
