A search for charged leptons with large impact parameters using $139\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ $pp$ 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 $340\text{ }\text{ }\mathrm{GeV}$, respectively, are excluded at 95% confidence level, drastically improving on the previous best limits from LEP.