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An excess of J/ψ yield at very low transverse momentum (pT<0.3 GeV/c), originating from coherent photoproduction, is observed in peripheral and semicentral hadronic Pb−Pb collisions at a center-of-mass energy per nucleon pair of sNN−−−√=5.02 TeV. The measurement is performed with the ALICE detector via the dimuon decay channel at forward rapidity (2.5<y<4). The nuclear modification factor at very low pT and the coherent photoproduction cross section are measured as a function of centrality down to the 10% most central collisions. These results extend the previous study at sNN−−−√=2.76 TeV, confirming the clear excess over hadronic production in the pT range 0−0.3 GeV/c and the centrality range 70−90%, and establishing an excess with a significance greater than 5σ also in the 50−70% and 30−50% centrality ranges. The results are compared with earlier measurements at sNN−−−√=2.76 TeV and with different theoretical predictions aiming at describing how coherent photoproduction occurs in hadronic interactions with nuclear overlap.