  The most massive galaxy clusters in the Universe host tens to hundreds of
massive satellite galaxies, but it is unclear if these satellites are able to
retain their own gaseous atmospheres. We analyze the evolution of $\sim90,000$
satellites of stellar mass $\sim10^{9-12.5}\,M_\odot$ around 352 galaxy
clusters of mass $M_{\rm vir}\sim10^{14.3-15.4}\,M_\odot$ at $z=0$ from the new
TNG-Cluster suite of cosmological magneto-hydrodynamical galaxy cluster
simulations. The number of satellites per host increases with host mass, and
the 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 this
fraction increases with stellar mass. Lower mass satellites
$\sim10^{9-10}\,M_\odot$ are more likely to retain part of their cold
interstellar medium, consistent with ram pressure preferentially removing hot
extended gas first. At higher stellar masses $\sim10^{10.5-12.5}\,M_\odot$ the
fraction of gas-rich satellites increases to unity, and nearly all satellites
retain a sizeable portion of their hot, spatially-extended circumgalactic
medium (CGM), despite the ejective activity of their supermassive black holes.
According to TNG-Cluster, the CGM of these gaseous satellites can be seen in
soft X-ray emission (0.5-2.0 keV) that is $\gtrsim10$ times brighter than the
local background. This X-ray surface brightness excess around satellites
extends to $\sim30-100$ kpc, and is strongest for galaxies with higher stellar
masses and larger host-centric distances. Approximately 10 per cent of the soft
X-ray emission in cluster outskirts $\sim0.75-1.5R_{\rm 200c}$ originates from
satellites. The CGM of member galaxies reflects the dynamics of
cluster-satellite interactions and contributes to the observationally-inferred
properties of the intracluster medium.

Astrophysics of Galaxies;Cosmology and Nongalactic Astrophysics