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The most massive galaxy clusters in the Universe host tens to hundreds ofmassive satellite galaxies, but it is unclear if these satellites are able toretain their own gaseous atmospheres. We analyze the evolution of $\sim90,000$satellites of stellar mass $\sim10^{9-12.5}\,M_\odot$ around 352 galaxyclusters of mass $M_{\rm vir}\sim10^{14.3-15.4}\,M_\odot$ at $z=0$ from the newTNG-Cluster suite of cosmological magneto-hydrodynamical galaxy clustersimulations. The number of satellites per host increases with host mass, andthe mass--richness relation broadly agrees with observations. A halo of mass$M_{\rm 200c}\sim10^{14.5}\,(10^{15})\,M_\odot$ hosts $\sim100\,(300)$satellites today. Only a minority of satellites retain some gas, and thisfraction increases with stellar mass. Lower mass satellites$\sim10^{9-10}\,M_\odot$ are more likely to retain part of their coldinterstellar medium, consistent with ram pressure preferentially removing hotextended gas first. At higher stellar masses $\sim10^{10.5-12.5}\,M_\odot$ thefraction of gas-rich satellites increases to unity, and nearly all satellitesretain a sizeable portion of their hot, spatially-extended circumgalacticmedium (CGM), despite the ejective activity of their supermassive black holes.According to TNG-Cluster, the CGM of these gaseous satellites can be seen insoft X-ray emission (0.5-2.0 keV) that is $\gtrsim10$ times brighter than thelocal background. This X-ray surface brightness excess around satellitesextends to $\sim30-100$ kpc, and is strongest for galaxies with higher stellarmasses and larger host-centric distances. Approximately 10 per cent of the softX-ray emission in cluster outskirts $\sim0.75-1.5R_{\rm 200c}$ originates fromsatellites. The CGM of member galaxies reflects the dynamics ofcluster-satellite interactions and contributes to the observationally-inferredproperties of the intracluster medium.

The hot circumgalactic media of massive cluster satellites in the TNG-Cluster simulation: existence and detectability