  The first study of $C\!P$ violation in the decay mode
$B^\pm\to[K^+K^-\pi^+\pi^-]_D h^\pm$, with $h=K,\pi$, is presented, exploiting
a data sample of proton-proton collisions collected by the LHCb experiment that
corresponds to an integrated luminosity of $9$ fb$^{-1}$. The analysis is
performed in bins of phase space, which are optimised for sensitivity to local
$C\!P$ asymmetries. $C\!P$-violating observables that are sensitive to the
angle $\gamma$ of the Unitarity Triangle are determined. The analysis requires
external information on charm-decay parameters, which are currently taken from
an amplitude analysis of LHCb data, but can be updated in the future when
direct measurements become available. Measurements are also performed of
phase-space integrated observables for $B^\pm\to[K^+K^-\pi^+\pi^-]_D h^\pm$ and
$B^\pm\to[\pi^+\pi^-\pi^+\pi^-]_D h^\pm$ decays.
