A search is presented for long-lived particles decaying into an oppositely charged lepton pair, μ + μ − , e + e − , or e ± μ ∓, that form a vertex within the inner tracking system of the ATLAS detector at the Large Hadron Collider, displaced from the primary proton–proton interaction region. The analysis uses the 140 fb − 1 of Run-2 data collected at s = 13 TeV by the ATLAS experiment in 2015–2018. The results of the analysis are interpreted in the context of three benchmark models covering masses from 0.1 to 2.2 TeV and a range of mean proper lifetimes times the speed of light from 1 to 10 000 mm. The first model is a generic Z′ boson pair-produced by a new heavy scalar, with the Z′ decaying into lepton pairs. The remaining two models are R-parity violating supersymmetric models in which the lightest neutralino χ ˜ 1 0 decays into ℓ + ℓ ′ − ν ( ℓ , ℓ ′ = e , μ). The models differ by the mode of production of the χ ˜ 1 0 , which can be produced via the decay of pairs of gluinos or of pairs of charginos and neutralinos ( χ ˜ 1 ± χ ˜ 1 0 , χ ˜ 1 ± χ ˜ 2 0 , or χ ˜ 2 0 χ ˜ 1 0 ). Although each benchmark sample includes pair-produced LLPs, only a single vertex is required to be reconstructed. No dilepton displaced vertex candidate is observed and the results are presented as upper limits on the production cross-sections. This analysis sets leading limits on the production cross-sections for multiple models, including parameter space that has never been directly probed.