Observing small-scale $\gamma$-ray anisotropies with the Cherenkov
  Telescope Array
  Disentangling the composition of the diffuse gamma-ray background (DGRB) is a
major challenge in gamma-ray astronomy. It is presumed that at the highest
energies, the DGRB is dominated by relatively few, still unresolved point
sources. This conjecture has recently been supported by the measurement of
small-scale anisotropies in the DGRB by the Fermi Large Area Telescope (LAT) up
to energies of 500 GeV. We show how such anisotropies can be searched for with
the forthcoming Earth-bound Cherenkov Telescope Array (CTA) up to the TeV
range. We investigate different observation modes to analyse CTA data for
small-scale anisotropies and propose the projected extragalactic large-area sky
survey as the most promising data set. Relying on an up-to-date model of the
performance of the southern CTA, we find that CTA will be able to probe
anisotropies in the DGRB from unresolved point sources at a relative amplitude
of $C_{\rm P}^I/I^2_{\rm DGRB}\gtrsim 4\times 10^{-3}\,{\rm sr}$ at energies
above 30 GeV and angular scales $\lesssim 1.5^{\circ}$. Such DGRB anisotropies
have not yet been ruled out by the Fermi-LAT. The proposed analysis would
primarily clarify the contribution from blazars and misaligned active galactic
nuclei to the very-high-energy regime of the DGRB, as well as provide insight
into dark matter annihilation in Galactic and extragalactic density structures.
Finally, it constitutes a measurement with complementary systematic
uncertainties compared to the Fermi-LAT.
