Hyperfine Structure of $n{P}_{1\mathrm{/}2}$ Rydberg States in ${}^{85}$Rb

We measure the hyperfine structure of $n{P}_{1\mathrm{/}2}$ 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 $n{P}_{1\mathrm{/}2}$ 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 $n{P}_{1\mathrm{/}2}$ atoms.