The centrality dependence of the mean charged-particle multiplicity as a function of pseudorapidity is measured in approximately 1 $$\upmu $$ b $$^{-1}$$ of proton–lead collisions at a nucleon–nucleon centre-of-mass energy of $$\sqrt{s_{_\text {NN}}} = 5.02$$ $$\text {TeV}$$ using the ATLAS detector at the Large Hadron Collider. Charged particles with absolute pseudorapidity less than 2.7 are reconstructed using the ATLAS pixel detector. The collision centrality is characterised by the total transverse energy measured in the Pb-going direction of the forward calorimeter. The charged-particle pseudorapidity distributions are found to vary strongly with centrality, with an increasing asymmetry between the proton-going and Pb-going directions as the collisions become more central. Three different estimations of the number of nucleons participating in the collision have been carried out using the Glauber model as well as two Glauber–Gribov inspired extensions to the Glauber model. Charged-particle multiplicities per participant pair are found to vary differently for these three models, highlighting the importance of including colour fluctuations in nucleon–nucleon collisions in the modelling of the initial state of collisions.