We report measurements of the frequency and temperature dependence of ferromagnetic resonance (FMR) for a 15-nm-thick yttrium iron garnet (YIG) film grown by off-axis sputtering. Although the FMR linewidth is narrow at room temperature (corresponding to a damping coefficient $\alpha$ = (9.0 $\pm$ 0.2) $\times 10^{-4}$), comparable to previous results for high-quality YIG films of similar thickness, the linewidth increases strongly at low temperatures, by a factor of almost 30. This increase cannot be explained as due to two-magnon scattering from defects at the sample interfaces. We argue that the increased low-temperature linewidth is due to impurity relaxation mechanisms that have been investigated previously in bulk YIG samples. We suggest that the low-temperature linewidth is a useful figure of merit to guide the optimization of thin-film growth protocols because it is a particularly sensitive indicator of impurities.