Lunar samples returned by the Apollo program have provided insights into numerous solar system processes. However, no samples were returned from the lunar farside, where one of the Moon's most geologically important features resides: the 2500-km-diameter South Pole-Aitken basin (SPA). Here, we explore the hypothesis that lunar troctolite 76535 was excavated by SPA. This hypothesis is motivated by the sample's low peak shock pressure (<6 GPa), its substantial depth of origin (45-65 km), and its ancient Ar-40/Ar-39 age of 4.25 Ga. We use hydrodynamic simulations of crater formation to show that for vertically incident impactors, SPA is the only known basin that can excavate material from the depth and shock pressure range relevant to 76535. The thermal history of 76535 also rules out excavation where a magma ocean was locally present. However, for the vertical impacts modeled, delivery of 76535 to the Apollo 17 site, where it was collected, requires a second impact event that preserved the sample's low shock state. An alternative interpretation of the SPA origin is that 76535 originates from the Serenitatis, Fecunditatis, or Australe basins, if the inferred origin depth of 76535 is in error by similar to 20 km, or its inferred peak shock pressure is in error by a factor of similar to 2. These basins could also be candidates for excavating 76535, if oblique impacts yield lower shock pressures of material excavated from the relevant depth. If troctolite 76535 is in fact a sample of SPA, we find that its 4.25 Ga excavation age and the density of large (100-300 km diameter) impact basins within and on the rim of SPA are consistent with the monotonically decaying Neukum (1983) chronology.