This paper reports a search for a light CP-odd resonance with a mass of $20\,\mathrm{GeV}$ to $90\,\mathrm{GeV}$ in $13\,\mathrm{TeV}$ proton-proton collision data with an integrated luminosity of $140\,\mathrm{fb}^{-1}$ collected with the ATLAS detector at the Large Hadron Collider. The analysis assumes the resonance is produced via gluon-gluon fusion and decays exclusively into a $\tau^{+}\tau^{-}$ pair which decays into a fully leptonic $\mu^{+}\nu_{\mu} \bar{\nu}_{\tau} e^{-} \bar{\nu}_{e} \nu_{\tau}$ or $e^{+}\nu_{e}\bar{\nu}_{\tau} \mu^-\bar{\nu}_{\mu}\nu_{\tau}$ 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 between $3.0\,\mathrm{pb}$ and $67.5\,\mathrm{pb}$ at $95\%$ confidence level are set on the cross-section for producing a CP-odd Higgs boson that decays into a $\tau^+\tau^-$ pair.