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 $gamma-2beta_s$ 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.
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