A measurement of dielectron production in proton-proton (pp) collisions at sqrt[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 m_{ee} and pair transverse momentum p_{T,ee} that was previously inaccessible at the LHC. The cross section for dielectron production is studied as a function of m_{ee}, p_{T,ee}, and event multiplicity dN_{ch}/dη. The expected dielectron rate from hadron decays, called hadronic cocktail, utilizes a parametrization of the measured η/π^{0} 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<m_{ee}<0.6 GeV/c^{2} and for p_{T,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 m_{ee} and p_{T,ee} are extracted and compared with calculations of dielectron production from hadronic bremsstrahlung and thermal radiation within a hadronic many-body approach.