A bstract The first measurement of the e + e − pair production at low lepton pair transverse momentum ( p T , ee ) and low invariant mass ( m ee ) in non-central Pb–Pb collisions at s NN = 5 . 02 TeV at the LHC is presented. The dielectron production is studied with the ALICE detector at midrapidity ( |η e | < 0 . 8) as a function of invariant mass (0.4 ≤ m ee < 2 . 7 GeV/ c 2 ) in the 50–70% and 70–90% centrality classes for p T , ee < 0.1 GeV/ c , and as a function of p T , ee in three m ee intervals in the most peripheral Pb–Pb collisions. Below a p T , ee of 0.1 GeV/ c , a clear excess of e + e − pairs is found compared to the expectations from known hadronic sources and predictions of thermal radiation from the medium. The m ee excess spectra are reproduced, within uncertainties, by different predictions of the photon–photon production of dielectrons, where the photons originate from the extremely strong electromagnetic fields generated by the highly Lorentz-contracted Pb nuclei. Lowest-order quantum electrodynamic (QED) calculations, as well as a model that takes into account the impact-parameter dependence of the average transverse momentum of the photons, also provide a good description of the p T , ee spectra. The measured p T , ee 2 of the excess p T , ee spectrum in peripheral Pb–Pb collisions is found to be comparable to the values observed previously at RHIC in a similar phase-space region.