  The most massive galaxy clusters in the Universe host hundreds of massive
satellite galaxies~$\mstar\sim10^{10-12.5} msun$, but it is unclear if these
satellites are able to retain their own gaseous halos. We analyze the evolution
of $\approx90,000$ satellites of stellar mass $\sim10^{9-12.5} msun$ around 352
galaxy clusters of mass $\mvir\sim10^{14.3-15.4} msun$ at $z=0$ from the new
TNG-Cluster suite of cosmological magneto-hydrodynamical galaxy cluster
simulations. The number of massive satellites per host increases with host
mass, and the mass--richness relation broadly agrees with observations. A halo
of mass $\mvirhost\sim10^{14.5} (10^{15}) msun$ hosts $\sim100 (300)$
satellites today. Only a minority of satellites retain some gas, hot or cold,
and this fraction increases with stellar mass. Lower-mass satellites
$\sim10^{9-10} msun$ 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 $\sim10^{10.5-12.5} msun$, the fraction of
gas-rich satellites increases to unity, and nearly all satellites retain a
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
$\approx30-100$ kpc, and is strongest for galaxies with higher stellar masses
and larger host-centric distances. Approximately $10$~percent of the soft X-ray
emission in cluster outskirts $\approx0.75-1.5\rvir$ 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.
