I feel like there should be more of an explanation, but it's pretty straightforward. A blackbody absorbing energy will increase in mass. The absolute amount of increase is pretty miniscule, but it is not zero.
Since you ask about an object that does not also radiate energy, a blackhole might be a decent analogy. So, does a blackhole increase in mass when photons fall into it? Sure. If it helps, you can imagine that a photon of sufficiently high energy can produce pairs of electrons (or other particles) that could subsequently fall into the blackhole... well, except the antiparticle that would be annihilated shortly after creation. Either way, it seems easier to imagine the scenario with particles that have a rest mass because it more closely corresponds to our quotidian experience.