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In the absence of a clear hint of dark matter (DM) signals in the GeV regime so far, heavy, $\mathcal{O}$(TeV) DM candidates are gradually earning more and more attention within the community. Among others, extra-dimensional \textit{brane-world} models may produce thermal DM candidates with masses up to 100 TeV. These heavy DM candidates could be detected with the next generation of very-high-energy gamma-ray observatories such as the Cherenkov Telescope Array (CTA). In this work, we study the sensitivity of CTA to branon DM via the observation of representative astrophysical DM targets, namely dwarf spheroidal galaxies. In particular, we focus on Draco and Sculptor, two well-known dwarfs visible from the Northern and Southern Hemisphere, respectively. For each of these targets, we simulated 300\,h of CTA observations and studied the sensitivity of both CTA-North and CTA-South to branon annihilations using the latest publicly available instrument response functions and most recent analysis tools. We computed annihilation cross section values needed to reach a $5σ$ detection as a function of the branon mass. Additionally, in the absence of a predicted DM signal, we obtained $2σ$ upper limits on the annihilation cross section. These limits lie $1.5-2$ orders of magnitude above the thermal relic cross section value, depending on the considered branon mass. Yet, CTA will allow to exclude a significant portion of the brane tension-mass parameter space in the $0.1-60$ TeV branon mass range, and up to tensions of $\sim 10$ TeV. More importantly, CTA will significantly enlarge the region already excluded by AMS and CMS, and will provide valuable complementary information to future SKA radio observations. We conclude that CTA will possess potential to constrain brane-world models and, more in general, TeV DM candidates.