As a fundamental parameter in magnetism, the phenomenological Gilbert damping constant a determines the performance of many spintronic devices. For most magnetic materials, a is treated as an isotropic parameter entering the Landau-Lifshitz-Gilbert equation. However, could the Gilbert damping be anisotropic? Although several theoretical approaches have suggested that anisotropic a could appear in single-crystalline bulk systems, experimental evidence of its existence is scarce. Here, we report the emergence of anisotropic magnetic damping by exploring a quasi-two-dimensional single-crystalline ferromagnetic metal/semiconductor interface-that is, a Fe/GaAs(001) heterojunction. The observed anisotropic damping shows twofold C-2v symmetry, which is expected from the interplay of interfacial Rashba and Dresselhaus spin-orbit interaction, and is manifested by the anisotropic density of states at the Fe/GaAs (001) interface. This discovery of anisotropic damping will enrich the understanding of magnetization relaxation mechanisms and can provide a route towards the search for anisotropic damping at other ferromagnetic metal/semiconductor interfaces.