  We compute the structure-dependent axial and vector form factors for the
radiative leptonic decays $D_s\to \ell\nu_\ell\gamma$, where $\ell$ is a
charged lepton, as functions of the energy of the photon in the rest frame of
the $D_s$ meson. The computation is performed using gauge-field configurations
with 2+1+1 sea-quark flavours generated by the European Twisted Mass
Collaboration and the results have been extrapolated to the continuum limit.
For the vector form factor we observe a very significant partial cancellation
between the contributions from the emission of the photon from the strange
quark and that from the charm quark. The results for the form factors are used
to test the reliability of various Anz\"atze based on single-pole dominance and
its extensions, and we present a simple parametrization of the form factors
which fits our data very well and which can be used in future phenomenological
analyses. Using the form factors we compute the differential decay rate and the
branching ratio for the process $D_s\to e\nu_e\gamma$ as a function of the
lower cut-off on the photon energy. With a cut-off of 10 MeV for example, we
find a branching ratio of Br$(E_\gamma>10\,\mathrm{MeV})=4.4(3)\times 10^{-6}$
which, unlike some model calculations, is consistent with the upper bound from
the BESIII experiment Br$(E_\gamma>10\,\mathrm{MeV})<1.3\times 10^{-4}$ at 90%
confidence level. Even for photon energies as low as 10 MeV, the decay $D_s\to
e\nu_e\gamma$ is dominated by the structure-dependent contribution to the
amplitude (unlike the decays with $\ell=\mu$ or $\tau$), confirming its value
in searches for hypothetical new physics as well as in determining the
Cabibbo-Kobayashi-Maskawa (CKM) parameters at $O(\alpha_\mathrm{em})$, where
$\alpha_{\mathrm{em}}$ is the fine-structure constant.
