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 $ {\mu}^{+}{\nu}_{\mu }{\overline{\nu}}_{\tau }{e}^{-}{\overline{\nu}}_e{\nu}_{\tau } $ or $ {e}^{+}{\nu}_e{\overline{\nu}}_{\tau }{\mu}^{-}{\overline{\nu}}_{\mu }{\nu}_{\tau } $ 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.[graphic not available: see fulltext]