The origin of the rest-optical emission of compact, red, high-redshift sources known as little red dots (LRDs) poses a major puzzle. If interpreted as starlight, it would imply that LRDs constitute the densest stellar systems in the Universe. However, alternative models suggest active galactic nuclei (AGN) may instead power the rest-optical continuum. Here, we present JWST/NIRSpec, NIRCam, and MIRI observations from the RUBIES and PRIMER programs of The Cliff: a bright LRD at z = 3.55 with an exceptional Balmer break, twice as strong as that of any high-redshift source previously observed. The spectra also reveal broad hydrogen (Hα FWHM ∼ 1500 km s−1) and He i emission, but no significant metal lines. We demonstrate that massive evolved stellar populations cannot explain the observed spectrum, even when considering unusually steep and strong dust attenuation or reasonable variations in the initial mass function. Moreover, the formally best-fit stellar mass and compact size (M∗ ∼ 1010.5 M, re ∼ 40 pc) would imply densities at which near-monthly stellar collisions might lead to significant X-ray emission. We argue that the Balmer break, emission lines, and Hα absorption line are instead most plausibly explained by a black hole star (BH*) scenario, in which dense gas surrounds a powerful ionising source. In contrast to recently proposed BH* models of dust-reddened AGN, we show that spectral fits in the rest UV to near-infrared favour an intrinsically redder continuum over strong dust reddening. This may point to a super-Eddington accreting massive black hole or, possibly, the presence of (super)massive stars in a nuclear star cluster. The Cliff is the clearest evidence to date that at least some LRDs are not ultra-dense massive galaxies, and are instead powered by a central ionising source embedded in dense, absorbing gas.