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We report measurements of the inclusive J/ψ yield and average transverse momentum as a function of charged-particle pseudorapidity density dNch/dη in p–Pb collisions at √sNN = 5.02 TeV with ALICE at the LHC. The observables are normalised to their corresponding averages in non-single diffractive events. An increase of the normalised J/ψ yield with normalised dNch/dη, measured at mid-rapidity, is observed at mid-rapidity and backward rapidity. At forward rapidity, a saturation of the relative yield is observed for high charged-particle multiplicities. The normalised average transverse momentum at forward and backward rapidities increases with multiplicity at low multiplicities and saturates beyond moderate multiplicities. In addition, the forward-to-backward nuclear modification factor ratio is also reported, showing an increasing suppression of J/ψ production at forward rapidity with respect to backward rapidity for increasing charged-particle multiplicity.