Spin-lattice relaxation of the nuclear spin system in p-type GaAs is studied using a three-stage experimental protocol including optical pumping and measuring the difference of the nuclear spin polarization before and after a dark interval of variable length. This method allows us to measure the spin-lattice relaxation time T1 of optically pumped nuclei "in the dark," that is, in the absence of illumination. The measured T1 values fall into the subsecond time range, being three orders of magnitude shorter than in earlier studied n-type GaAs. The drastic difference is further emphasized by magnetic-field and temperature dependencies of T1 in p-GaAs, showing no similarity to those in n-GaAs. This unexpected behavior finds its explanation in the spatial selectivity of the optical pumping in p-GaAs, that is only efficient in the vicinity of shallow donors, together with the quadrupole relaxation of nuclear spins, which is induced by electric fields within closely spaced donor-acceptor pairs. The developed theoretical model explains the whole set of experimental results.