  Liquid xenon (LXe) is a well-studied detector medium to search for rare
events in dark matter and neutrino physics. Two-phase xenon time projection
chambers (TPCs) can detect electronic and nuclear recoils with energy down to
kilo-electron volts (keV). In this paper, we characterize the response of a
single-phase liquid xenon proportional scintillation counter (LXePSC), which
produces electroluminescence directly in the liquid, to detect electronic
recoils at low energies. Our design uses a thin (10 - 25 $\mu$m diameter),
central anode wire in a cylindrical LXe target where ionization electrons,
created from radiation particles, drift radially towards the anode, and
electroluminescence is produced. Both the primary scintillation (S1) and
electroluminescence (S2) are detected by photomultiplier tubes (PMTs)
surrounding the LXe target. Up to 17 photons are produced per electron,
obtained with a 10 $\mu$m diameter anode wire, allowing for the highly
efficient detection of electronic recoils from beta decays of a tritium source
down to roughly 1 keV. Single electrons, from photo-emission of the cathode
wires, are observed at a gain of 1.8 photoelectrons (PE) per electron. The
delayed signals following the S2 signals are dominated by single-photon-like
hits, without evidence for electron signals observed in the two-phase xenon
TPCs. We discuss the potential application of such a LXePSC for reactor
neutrino detection via Coherent Elastic Neutrino Nucleus Scattering
(CE$\nu$NS).
