Suppose you have some form of blackbody, and surround it with a perfect (or even just pretty good) spherical mirror, and coat the outside surface of the mirror in something (say clay, and then an insulator). Suppose they are both at 50 degrees C, and in a vacuum. Both the mirror and the blackbody emit infrared radiation, correct? But the mirror will reflect most incoming light, while the blackbody will absorb it. It seems like the mirror will get colder, while the blackbody will get warmer. That sounds like a violation of thermodynamics, though - so what would actually happen?
The amount of energy emitted as thermal radiation from a body is proportional to the fraction of radiated energy it absorbs. The reflective surface of a perfect mirror, therefore, does not emit any thermal radiation.
This means that in the setup you described the vacuum+perfect mirror effectively cut your system into 2 parts with no heat transferred between them. The blackbody will emit thermal radiation into the cavity until the radiation in the cavity has the same temperature as the blackbody, at which point equilibrium will have been reached. No energy is emitted or absorbed by the mirrored surface, however, so this is totally isolated from the second part of the system, consisting of the mirrors coating and surroundings, which will reach there own equilibrium independent of the blackbody. (If the mirror's surroundings consist of an infinite empty vacuum then that can be considered a heat reservoir at absolute zero temperature)