  We study Nevanlinna theory of meromorphic mappings from a geodesic ball of a
general complete K\"ahler manifold with non-negative Ricci curvature into a
complex projective manifold by introducing a heat kernel method. When dimension
of a target manifold is not greater than one of a source manifold, we establish
a second main theorem which is a generalization of the classical second main
theorem for a ball of $\mathbb C^m.$ If a source manifold is non-compact and it
carries a positive global Green function, then we establish a global second
main theorem for the source manifold. As a result, we obtain a Picard's theorem
for complete K\"ahler manifolds with non-negative Ricci curvature.
