We report on the discovery of strong tidal features around a dwarf spheroidal galaxy in the Hydra I galaxy cluster, indicating its ongoing tidal disruption. This very low surface brightness object, HCC-087, was originally classified as an early-type dwarf in the Hydra Cluster Catalogue (HCC), but our re-analysis of the ESO-VLT/FORS images of the HCC unearthed a clear indication of an S-shaped morphology and a large spatial extent. Its shape, luminosity (MV = –11.6 mag), and physical size (at a half-light radius of 3.1 kpc and a full length of ~5.9 kpc) are comparable to the recently discovered NGC 4449B and the Sagittarius dwarf spheroidal, all of which are undergoing clear tidal disruption. Aided by N-body simulations we argue that HCC-087 is currently at its first apocenter, at 150 kpc, around the cluster center and that it is being tidally disrupted by the galaxy cluster's potential itself. An interaction with the nearby (50 kpc) S0 cluster galaxy HCC-005, at M * ~ 3 × 1010 M ☉ is rather unlikely, as this constellation requires a significant amount of dynamical friction and thus low relative velocities. The S-shaped morphology and large spatial extent of the satellite would, however, also appear if HCC-087 would orbit the cluster center. These features appear to be characteristic properties of satellites that are seen in the process of being tidally disrupted, independent of the environment of the destruction. An important finding of our simulations is an orientation of the tidal tails perpendicular to the orbit.