[en] In this work, we extend for the first time the spherically symmetric Schwarzschild and Schwarzschild–De Sitter solutions with a Finsler–Randers-type perturbation which is generated by a covector A${}_{\mathrm{\gamma <!--\; ??\; -->}}$. This gives a locally anisotropic character to the metric and induces a deviation from the Riemannian models of gravity. A natural framework for this study is the Lorentz tangent bundle of a spacetime manifold. We apply the generalized field equations to the perturbed metric and derive the dynamics for the covector A${}_{\mathrm{\gamma <!--\; ??\; -->}}$. Finally, we find the timelike, spacelike and null paths on the Schwarzschild–Randers spacetime, we solve the timelike ones numerically and we compare them with the classic geodesics of general relativity. The obtained solutions are new and they enrich the corresponding literature.