We present a measurement of azimuthal correlations between inclusive J/ψ and charged hadrons in p–Pb collisions recorded with the ALICE detector at the CERN LHC. The J/ψ are reconstructed at forward (p-going, 2.03<y<3.53) and backward (Pb-going, −4.46<y<−2.96) rapidity via their μ⁺μ⁻ decay channel, while the charged hadrons are reconstructed at mid-rapidity (|η|<1.8). The correlations are expressed in terms of associated charged-hadron yields per J/ψ trigger. A rapidity gap of at least 1.5 units is required between the trigger J/ψ and the associated charged hadrons. Possible correlations due to collective effects are assessed by subtracting the associated per-trigger yields in the low-multiplicity collisions from those in the high-multiplicity collisions. After the subtraction, we observe a strong indication of remaining symmetric structures at Δφ≈0 and Δφ≈π similar to those previously found in two-particle correlations at middle and forward rapidity. The corresponding second-order Fourier coefficient (v₂) in the transverse momentum interval between 3 and 6 GeV/c is found to be positive with a significance of about 5σ. The obtained results are similar to the J/ψ v₂ coefficients measured in Pb–Pb collisions at s_NN=5.02 TeV, suggesting a common mechanism at the origin of the J/ψ v₂.