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The first study of ϕ-meson production in p–Pb collisions at forward and backward rapidity, at a nucleon–nucleon centre-of-mass energy √sNN=5.02 TeV, has been performed with the ALICE apparatus at the LHC. The ϕ-mesons have been identified in the dimuon decay channel in the transverse momentum (pT) range 1<pT<7 GeV/c, both in the p-going (2.03<y<3.53) and the Pb-going (−4.46<y<−2.96) directions — where y stands for the rapidity in the nucleon–nucleon centre-of-mass — the integrated luminosity amounting to 5.01±0.19 nb−1 and 5.81±0.20 nb−1, respectively, for the two data samples. Differential cross sections as a function of transverse momentum and rapidity are presented. The forward–backward ratio for ϕ-meson production is measured for 2.96<|y|<3.53, resulting in a ratio ∼0.5 with no significant pT dependence within the uncertainties. The pT dependence of the ϕ nuclear modification factor RpPb exhibits an enhancement up to a factor 1.6 at pT=3–4 GeV/c in the Pb-going direction. The pT dependence of the ϕ-meson cross section in pp collisions at √s=2.76 TeV, which is used to determine a reference for the p–Pb results, is also presented here for 1<pT<5 GeV/c and 2.5<y<4, for a 78±3 nb−1 integrated luminosity sample.