We measure the hyperfine structure of
nP1/2 Rydberg states for
n= 42, 43, 44, and 46 using mm-wave spectroscopy on an ensemble of laser-cooled
85Rb atoms. Systematic uncertainties in our measurement from the Zeeman splittings induced by stray magnetic fields and dipole-dipole interactions between two Rydberg atoms are factored in with the obtained statistical uncertainty. Our final measurement of the
nP1/2 hyperfine coupling constant is
AHFS=1.443(31) GHz. This measurement is useful for studies of long-range Rydberg molecules, Rydberg electrometry, and quantum simulation with dipole-dipole interactions involving
nP1/2 atoms.