One of the fundamental challenges for future leptonic colliders and neutrino factories as well as for high-sensitivity studies of lepton universality is to design and construct new high-intensity sources of muons and positrons. The next-generation sources should increase the intensity of the presently operating ones by at least 3 orders of magnitude and include an important option of producing longitudinally polarized leptons. The main effort to achieve this goal has been focused, so far, on the proton-beam-driven muon sources and electron-beam-driven positron sources. In this paper, we present exploratory studies of an alternative scheme which is based on high-intensity megawatt-class photon beams. Such beams could be delivered in the future by the Gamma Factory (GF) project. One of the GF multiple goals is to increase the energy range (by more than one order of magnitude) and the intensity (by more than 6 orders of magnitude) of presently operating photon sources. Such a leap can be achieved by extending the present hadron-collider modus operandi of the LHC with a new GF-operation-mode, allowing the collisions of beams with laser pulses. The exploratory studies presented in this paper demonstrate that more than 10 muons of both signs and more than 10 electrons/positrons per second can be produced by a GF source.