A search for charged leptons with large impact parameters using 139 fb^{-1} of sqrt[s]=13 TeV pp collision data from the ATLAS detector at the LHC is presented, addressing a long-standing gap in coverage of possible new physics signatures. Results are consistent with the background prediction. This search provides unique sensitivity to long-lived scalar supersymmetric lepton partners (sleptons). For lifetimes of 0.1 ns, selectron, smuon, and stau masses up to 720, 680, and 340 GeV, respectively, are excluded at 95% confidence level, drastically improving on the previous best limits from LEP.