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 α=(9.0±0.2)×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 point out that the increased low-temperature linewidth can be explained by impurity relaxation mechanisms that were elucidated 50 years ago in bulk YIG samples. High-purity starting materials and careful optimization of growth protocols to avoid nonstoichiometries should therefore be employed for making low-temperature thin-film YIG devices. © 2017 American Physical Society.