  For the first time, the production of a massive dark photon (DP) in the
$\gamma e^-$ scattering at the future lepton colliders ILC, CLIC, and CEPC is
examined. The invisible decay mode of the DP is addressed. We have studied both
the unpolarized scattering and the collision of the Compton backscattered
photons with the polarized electron. The missing energy distributions are
shown. We have considered the wide range 10 GeV -- 1000 GeV of the DP mass
$m_{A'}$. The excluded regions at the 95\% C.L. in the plane $(\varepsilon,
m_{A'})$, where $\varepsilon$ is the kinetic mixing parameter, are obtained.
