  The first measurement of the ${\rm e}^{+}{\rm e}^{-}$ pair production at low
lepton pair transverse momentum ($p_{\rm T,ee}$) and low invariant mass
($m_{\rm ee}$) in non-central Pb$-$Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$
TeV at the LHC is presented. The dielectron production is studied with the
ALICE detector at midrapidity ($|\eta_{\rm e}| < 0.8$) as a function of
invariant mass ($0.4 \leq m_{\rm ee} < 2.7$ GeV/$c^2$) in the 50$-$70% and
70$-$90% centrality classes for $p_{\rm T,ee} < 0.1$ GeV/$c$, and as a function
of $p_{\rm T,ee}$ in three $m_{\rm ee}$ intervals in the most peripheral
Pb$-$Pb collisions. Below a $p_{\rm T,ee}$ of 0.1 GeV/$c$, a clear excess of
${\rm e}^{+}{\rm e}^{-}$ pairs is found compared to the expectations from known
hadronic sources and predictions of thermal radiation from the medium. The
$m_{\rm ee}$ 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 $p_{\rm T,ee}$ spectra. The measured $\sqrt{\langle
p_{\rm T,ee}^{2} \rangle}$ of the excess $p_{\rm T,ee}$ spectrum in peripheral
Pb$-$Pb collisions is found to be comparable to the values observed previously
at RHIC in a similar phase-space region.
