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 $|\eta|<2.3$ and transverse momentum between $0.1$ GeV and $22$ GeV in a dataset corresponding to an integrated luminosity of $1$ $\mu b^{-1}$. 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 $-4.9<\eta<-3.1$, which is in the direction of the outgoing lead beam. Three different estimations of the number of nucleons participating in the $p+$Pb collision are carried out using the Glauber model and two Glauber-Gribov colour-fluctuation extensions to the Glauber model. The values of the nuclear modification factors are found to vary significantly as a function of rapidity and transverse momentum. A broad peak is observed for all centralities and rapidities in the nuclear modification factors for charged-particle transverse momentum values around $3$ GeV. The magnitude of the peak increases for more central collisions as well as rapidity ranges closer to the direction of the outgoing lead nucleus.
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