Seismic observations show the Earth's inner core has significant and unexplained variation in seismic attenuation with position, depth and direction. Interpreting these observations is difficult without knowledge of the visco- or anelastic dissipation processes active in iron under inner core conditions. Here, a previously unconsidered attenuation mechanism is observed in zinc, a low pressure analog of hcp-iron, during small strain sinusoidal deformation experiments. The experiments were performed in a deformation-DIA combined with X-radiography, at seismic frequencies (∼0.003–0.1 Hz), high pressure and temperatures up to ∼80% of melting temperature. Significant dissipation (0.077 ≤ Q⁻¹(ω) ≤ 0.488) is observed along with frequency dependent softening of zinc's Young's modulus and an extremely small activation energy for creep (⩽7 kJ mol⁻¹). In addition, during sinusoidal deformation the original microstructure is replaced by one with a reduced dislocation density and small, uniform, grain size. This combination of behavior collectively reflects a mode of deformation called “internal stress superplasticity”; this deformation mechanism is unique to anisotropic materials and activated by cyclic loading generating large internal stresses. Here we observe a new form of internal stress superplasticity, which we name as “elastic strain mismatch superplasticity.” In it the large stresses are caused by the compressional anisotropy. If this mechanism is also active in hcp-iron and the Earth's inner-core it will be a contributor to inner-core observed seismic attenuation and constrain the maximum inner-core grain-size to ≲10 km.