During dynamic nuclear polarization (DNP) at 1.5 K and 5 T, (129)Xe nuclear magnetic resonance (NMR) spectra of a homogeneous xenon/1-propanol/trityl-radical solid mixture exhibit a single peak, broadened by (1)H neighbors. A second peak appears upon annealing for several hours at 125 K. Its characteristic width and chemical shift indicate the presence of spontaneously formed pure Xe clusters. Microwave irradiation at the appropriate frequencies can bring both peaks to either positive or negative polarization. The peculiar time evolution of (129)Xe polarization in pure Xe clusters during DNP can be modelled as an interplay of spin diffusion and T(1) relaxation. Our simple spherical-cluster model offers a sensitive tool to evaluate major DNP parameters in situ, revealing a severe spin-diffusion bottleneck at the cluster boundaries and a significant sample overheating due to microwave irradiation. Subsequent DNP system modifications designed to reduce the overheating resulted in four-fold increase of (129)Xe polarization, from 5.3% to 21%.