We study a bino-like light neutralino ((chi) over tilde (0)(1)) produced at the LHC from the decay of a scalar lepton ((e) over tilde (L)) through the process pp -> (e) over tilde (L) -> e (chi) over tilde (0)(1) in the context of R-parity-violating (RPV) supersymmetry where (chi) over tilde (0)(1) is the lightest supersymmetric particle. For small masses and RPV couplings, the neutralino is naturally long-lived and its decay products can be identified as displaced tracks. Following existing searches, we propose a displaced-vertex search strategy for such a light neutralino with a single RPV coupling switched on, lambda(111)', in the mass range 10 GeV less than or similar to m((chi) over tilde 10) less than or similar to 230 GeV. We perform Monte Carlo simulations and conclude that at the high-luminosity LHC, the proposed search can probe values of lambda(111)' down to two orders of magnitude smaller than current bounds and up to 40 times smaller than projected limits from monolepton searches.