This paper reports a search for a light CP-odd scalar resonance with a mass of 20 GeV to 90 GeV in 13 TeV proton-proton collision data with an integrated luminosity of 140 fb⁻¹ collected with the ATLAS detector at the Large Hadron Collider. The analysis assumes the resonance is produced via gluon-gluon fusion and decays into a τ⁺τ⁻ pair which subsequently decays into a fully leptonic μ⁺ννe⁻νν or e⁺ννμ⁻νν final state. No significant excess of events above the predicted Standard Model background is observed. The results are interpreted within a flavour-aligned two-Higgs-doublet model, and a model-independent cross-section interpretation is also given. Upper limits at 95 confidence level between 3.0 pb and 68 pb are set on the cross-section for producing a CP-odd Higgs boson that decays into a τ⁺τ⁻ pair.