The spin relaxation in undoped InSb films grown on GaAs has been investigated in the temperature range from 77 to 290 K. Two distinct lifetime values have been extracted, 1 and 2.5 ps, dependent on film thickness. Comparison of this data with a multilayer transport analysis of the films suggests that the longer time (~2.5 ps at 290 K) is associated with the central intrinsic region of the film, while the shorter time (~1 ps) is related to the highly dislocated accumulation region at the film-substrate interface. Whereas previous work on InAs films grown on GaAs showed that the native surface defect resulted in an additional charge accumulation layer with high conductivity but very short spin lifetime, in InSb layers the surface states introduce a depletion region. We infer that InSb could be a more attractive candidate for spintronic applications than InAs.