  This paper presents results of searches for electroweak production of
supersymmetric particles in models with compressed mass spectra. The searches
use 139 fb$^{-1}$ of $\sqrt{s}=13$ TeV proton-proton collision data collected
by the ATLAS experiment at the Large Hadron Collider. Events with missing
transverse momentum and two same-flavor, oppositely charged, low transverse
momentum leptons are selected, and are further categorized by the presence of
hadronic activity from initial-state radiation or a topology compatible with
vector-boson fusion processes. The data are found to be consistent with
predictions from the Standard Model. The results are interpreted using
simplified models of $R$-parity-conserving supersymmetry in which the lightest
supersymmetric partner is a neutralino with a mass similar to the lightest
chargino, the second-to-lightest neutralino or the slepton. Lower limits on the
masses of charginos in different simplified models range from 193 GeV to 240
GeV for moderate mass splittings, and extend down to mass splittings of 1.5 GeV
to 2.4 GeV at the LEP chargino bounds (92.4 GeV). Similar lower limits on
degenerate light-flavor sleptons extend up to masses of 251 GeV and down to
mass splittings of 550 MeV. Constraints on vector-boson fusion production of
electroweak SUSY states are also presented.
