Measurements of the per-event charged-particle yield as a function of the charged-particle transverse momentum and rapidity are performed using $p+$Pb collision data collected by the ATLAS experiment at the LHC at a centre-of-mass energy of $\sqrt{s_{NN}}=5.02$ TeV. Charged particles are reconstructed over pseudorapidity $|η|<2.3$ and transverse momentum between $0.1$ GeV and $22$ GeV in a dataset corresponding to an integrated luminosity of $1$ $μb^{-1}$. The results are presented in the form of charged-particle nuclear modification factors, where the $p+$Pb charged-particle multiplicities are compared between central and peripheral $p+$Pb collisions as well as to charged-particle cross sections measured in pp collisions. The $p+$Pb collision centrality is characterized by the total transverse energy measured in $-4.9