Anelastic materials exhibit gradual full recovery of deformation once a load is removed, leading to dissipation of internal mechanical energy. As a consequence, anelastic materials are being investigated for mechanical damping applications. At the macroscopic scale, however, anelasticity is usually very small or negligible, especially in single-crystalline materials. Here, we show that single-crystalline ZnO and p-doped Si nanowires can exhibit anelastic behaviour that is up to four orders of magnitude larger than the largest anelasticity observed in bulk materials, with a timescale on the order of minutes. In situ scanning electron microscope tests of individual nanowires showed that, on removal of the bending load and instantaneous recovery of the elastic strain, a substantial portion of the total strain gradually recovers with time. We attribute this large anelasticity to stress-gradient-induced migration of point defects, as supported by electron energy loss spectroscopy measurements and also by the fact that no anelastic behaviour could be observed under tension. We model this behaviour through a theoretical framework by point defect diffusion under a high strain gradient and short diffusion distance, expanding the classic Gorsky theory. Finally, we show that ZnO single-crystalline nanowires exhibit a high damping merit index, suggesting that crystalline nanowires with point defects are promising materials for energy damping applications.