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We prove that there exist solutions for a nonparametric capillary problem in a wide class of Riemannian manifolds endowed with a Killing vector field. In other terms, we prove the existence of Killing graphs with prescribed mean curvature and prescribed contact angle along its boundary. These results may be useful for modeling stationary hypersurfaces under the influence of a nonhomogeneous gravitational field defined over an arbitrary Riemannian manifold.