The answer to your question is that in theory yes, but in practice, no. All the mirrors we can make are not 100% reflective; they still absorb some light. The sphere would also need to contain a perfect vacuum inside, which means no air inside. The oxygen and nitrogen molecules in air can absorb light and convert it into heat.
Setting up an experiment where we could create a perfect sphere, with a perfect mirror inside, then also introduce light without letting air would be basically impossible since we cannot create a perfect sphere nor absolute vacuum. In the hypothetical scenario where we could do all these things, then from a classical perspective the light should bounce around forever inside. There may be some relativistic or quantum effects that still cause the light to stop bouncing around, but I don't know enough about those topics to explain anything more.
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