The most massive galaxy clusters in the Universe host tens to hundreds of massive satellite galaxies M⋆ ∼ 1010 − 12.5 M⊙, but it is unclear if these satellites are able to retain their own gaseous atmospheres. We analyze the evolution of ≈90 000 satellites of stellar mass ∼109 − 12.5 M⊙ around 352 galaxy clusters of mass M200c ∼ 1014.3 − 15.4 M⊙ 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 M200chost ∼ 1014.5(1015) M⊙ hosts ∼100 (300) satellites today. Only a minority of satellites retain some gas, hot or cold, and this fraction increases with stellar mass. lower-mass satellites ∼109 − 10 M⊙ 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 ∼1010.5 − 12.5 M⊙, 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, ≳10 times brighter than the local background. This X-ray surface brightness excess around satellites extends to ≈30 − 100 kpc, and is strongest for galaxies with higher stellar masses and larger host-centric distances. Approximately 10% of the soft X-ray emission in cluster outskirts ≈0.75 − 1.5 R200c 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.