  The centrality dependence of the mean charged-particle multiplicity as a
function of pseudorapidity is measured in approximately 1 $\mu$b$^{-1}$ of
proton--lead collisions at a nucleon--nucleon centre-of-mass energy of
$\sqrt{s_{_{\rm{NN}}}} = 5.02$ 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 $p$+Pb 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 $p$+Pb 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 $p$+Pb collisions.
