We examine the possibility to detect new SM-neutral vector bosons (Z') that couple exclusively to leptons in the electron-positron mode of the Future Circular Collider (FCC-ee). Focusing on the Z' production with a radiated photon search channel, we show that the FCC-ee can significantly extend the unprobed parameter space by increasing the exclusion in the coupling by one to two orders of magnitude in the kinematically allowed mass range (from 10 to 365 GeV), with the leading sensitivity being driven by the muon channel. In doing so, it outperforms other proposed lepton collider options such as CLIC and ILC in this range of masses. Further, we discuss the possibility of improving the sensitivity of the FCC-ee to this model through the modification of the dilepton invariant mass resolution and the photon energy resolution. The impact of systematic uncertainties on the expected sensitivities is also studied.