One of the fundamental challenges for the future leptonic colliders and neutrino factories as well as for the 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 three orders of the magnitude and include an important option of producing longitudinally polarised 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 six orders of magnitude) of the presently operating photon sources. Such a leap can be achieved by extending the present hadron-collider modus operandi of the LHC with the new GF-operation-mode, allowing to collide atomic beams with laser pulses. The exploratory studies presented in this paper demonstrate that more than 10$^{13}$ muons of both signs and more than 10$^{16}$ electrons/positrons per second can be produced by the GF source.