The thermal and mechanical evolution of Europa and comparable icy satellites-the physics behind creating and sustaining a subsurface water ocean-depends almost entirely on the mechanical dissipation of tidal energy in ice to produce heat, the mechanism(s) of which remain poorly understood. In deformation experiments, we combine steady-state creep and low-frequency, small-strain periodic loading, similar conditions in which tectonics and tidal flexing are occurring simultaneously. The data reveal that the relevant, power-law attenuation in ice (i) is non-linear, depending on strain amplitude, (ii) is independent of grain size, and (iii) exceeds in absorption the prediction of the Maxwell solid model by an order of magnitude. The Maxwell solid model is widely used to model the dynamics of planetary ice shells, so this discrepancy is important. The prevalent understanding of damping in the geophysical context is that it is controlled by chemical diffusion on grain boundaries, which renders attenuation strongly dependent on grain size. In sharp contrast, our results indicate instead the importance of intracrystalline dislocations and their spatial interactions as the critical structural variable affecting dissipation. These dislocation structures are controlled by stress and realized by accumulated plastic strain. Thus, tectonics and attenuation are coupled, which, beyond the icy satellite/subsurface ocean problem, has implications also for understanding the attenuation of seismic waves in deforming regions of the Earth's upper mantle.