  We measure the hyperfine structure of $nP_{1/2}$ Rydberg states using mm-wave
spectroscopy on an ensemble of laser-cooled $^{85}$Rb 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 $nP_{1/2}$ hyperfine coupling constant is $A_{\text{hfs}}=1.443(31)$~GHz.
This measurement is useful for studies of long-range Rydberg molecules, Rydberg
electrometry, and quantum simulation with dipole-dipole interactions involving
$nP_{1/2}$ atoms.
