The results of measuring the capacity of a thin-film solid-state lithium-ion battery (SSLIB) Ti|Si@O@Al|LiPON|LiCoO2|Ti at different charge currents are reported. It is shown that the dependence of the capacity on the current density follows the Peukert's law, which is characterized by a low value of the Peukert’s exponent in the region of low currents and a high value of the exponent in the region of high currents. A capacity model for a SSLIB based on the balance of diffusion and migration currents is proposed and the dependence of the capacity on the charge current is derived. The obtained dependence is well approximated by Peukert's law with an exponent of 1.39, which is intermediate between the exponents for lithium-ion batteries with liquid electrolyte and SSLIBs.