A measurement of dielectron production in proton-proton (pp) collisions at s=13 TeV, recorded with the ALICE detector at the CERN LHC, is presented in this Letter. The data set was recorded with a reduced magnetic solenoid field. This enables the investigation of a kinematic domain at low dielectron (ee) invariant mass mee and pair transverse momentum pT,ee that was previously inaccessible at the LHC. The cross section for dielectron production is studied as a function of mee, pT,ee, and event multiplicity dNch/dη. The expected dielectron rate from hadron decays, called hadronic cocktail, utilizes a parametrization of the measured η/π0 ratio in pp and proton-nucleus collisions, assuming that this ratio shows no strong dependence on collision energy at low transverse momentum. Comparison of the measured dielectron yield to the hadronic cocktail at 0.15<mee<0.6 GeV/c2 and for pT,ee<0.4 GeV/c indicates an enhancement of soft dielectrons, reminiscent of the "anomalous"soft-photon and soft-dilepton excess in hadron-hadron collisions reported by several experiments under different experimental conditions. The enhancement factor over the hadronic cocktail amounts to 1.61±0.13(stat)±0.17(syst,data)±0.34(syst,cocktail) in the ALICE acceptance. Acceptance-corrected excess spectra in mee and pT,ee are extracted and compared with calculations of dielectron production from hadronic bremsstrahlung and thermal radiation within a hadronic many-body approach.