  Collider, direct and indirect searches for dark matter have typically little
or no sensitivity to weakly interacting massive particles (WIMPs) with masses
above a few TeV. This rather unexplored regime can however be probed through
the search for distinctive gamma-ray spectral features produced by the
annihilation of WIMPs at very high energies. Here we present a dedicated search
for gamma-ray boxes -- sharp spectral features that cannot be mimicked by
astrophysical sources -- with the upcoming Cherenkov Telescope Array (CTA).
Using realistic projections for the instrument performance and detailed
background modelling, a profile likelihood analysis is implemented to derive
the expected upper limits and sensitivity reach after 100 h of observations
towards a $2^\circ\times2^\circ$ region around the Galactic centre. Our results
show that CTA will be able to probe gamma-ray boxes down to annihilation cross
sections of $10^{-27}-10^{-26}\,\text{cm}^3\text{/s}$ up to tens of TeV. We
also identify a number of concrete particle physics models providing thermal
dark matter candidates that can be used as target benchmarks in future search
campaigns. This constitutes a golden opportunity for CTA to either discover or
rule out multi-TeV thermal dark matter in a corner of parameter space where all
other experimental efforts are basically insensitive.
