  The CKM angle $\gamma$ is measured for the first time from mixing-induced
$CP$ violation between $B^0_s \rightarrow D_s^\mp K^\pm \pi^\pm \pi^\mp$ and
$\bar{B}^0_s \rightarrow D_s^\pm K^\mp \pi^\mp \pi^\pm$ decays reconstructed in
proton-proton collision data corresponding to an integrated luminosity of 9
${\rm fb}^{-1}$ recorded with the LHCb detector. A time-dependent amplitude
analysis is performed to extract the $CP$-violating weak phase
$\gamma-2\beta_s$ and, subsequently, $\gamma$ by taking the
$B^0_s$-$\bar{B}^0_s$ mixing phase $\beta_{s}$ as an external input. The
measurement yields $\gamma = (44 \pm 12)^\circ$ modulo $180^\circ$, where
statistical and systematic uncertainties are combined. An alternative
model-independent measurement, integrating over the five-dimensional phase
space of the decay, yields $\gamma = (44^{\,+\,20}_{\,-\,13})^\circ$ modulo
$180^\circ$. Moreover, the $B^0_s$-$\bar{B}^0_s$ oscillation frequency is
measured from the flavour-specific control channel $B^0_s \rightarrow D_s^-
\pi^+ \pi^+ \pi^-$ to be $\Delta m_s = (17.757 \pm 0.007 \,({\rm stat.}) \pm
0.008 \,({\rm syst.})) \text{ps}^{-1}$, consistent with and more precise than
the current world-average value.
