  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 three 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 six 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$^{13}$ muons of both signs and more than 10$^{16}$
electrons/positrons per second can be produced by a GF source.
