The development of high brightness electron sources can enable an increase in performance and reduction in size of extreme X-ray sources such as free electron lasers (FELs). A promising path to high brightness is through larger electric fields in radio-frequency (rf) photoinjectors. Recent experiments with 11.4 GHz copper accelerating cavities at cryogenic temperatures have demonstrated $500\text{ }\text{ }\mathrm{MV}/\mathrm{m}$ surface electric fields with low rf breakdown rates. However, when the surface electric fields are larger than $300\text{ }\text{ }\mathrm{MV}/\mathrm{m}$, the measured cavity quality factor, ${Q}_{0}$, decreases during the input rf pulse by up to 30%, recovering before the next rf pulse. In this paper, we present an experimental study of the rf losses, manifested as degradation of ${Q}_{0}$, in a copper cavity operated at cryogenic temperatures and high gradients. The experimental conditions range from temperatures of 10--77 K and rf pulse lengths of 100--800 ns, using surface electric fields up to $400\text{ }\text{ }\mathrm{MV}/\mathrm{m}$. We developed a model for the change in ${Q}_{0}$ using measured field emission currents and rf signals. We find that the ${Q}_{0}$ degradation is consistent with the rf power being absorbed by strong field emission currents accelerated inside the cavity.