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In this work, we investigate the anelastic deformation behavior of periodic three-dimensional (3D) nanolattices with extremely thin shell thicknesses using nanoindentation. The results show that the nanolattice continues to deform with time under a constant load. In the case of 30-nm-thick aluminum oxide nanolattices, the anelastic deformation accounts for up to 18.1% of the elastic deformation for a constant load of 500 μN. The nanolattices also exhibit up to 15.7% recovery after unloading. Finite element analysis (FEA) coupled with diffusion of point defects is conducted, which is in qualitative agreement with the experimental results. The anelastic behavior can be attributed to the diffusion of point defects in the presence of a stress gradient and is reversible when the deformation is removed. The FEA model quantifies the evolution of the stress gradient and defect concentration and demonstrates the important role of a wavy tube profile in the diffusion of point defects. The reported anelastic deformation behavior can shed light on time-dependent response of nanolattice materials with implication for energy dissipation applications.