One way to work around the heat dissipation issues in space (and also on earth) is to move to computing systems that operate entirely at cryogenic temperatures to take advantage of superconducting circuitry.

I've heard stories that over a decade ago teams inside hyperscalars had calculated that running completely cryogenically cooled data centers would be vastly cheaper than what we do now due to savings on resistive losses and the cost of eliminating waste heat. You don't have to get rid of heat that you don't generate in the first place.

The issue is that at the moment there are very few IC components and processes that have been engineered to run at cryogenic temperatures. Replicating the entirety of the existing data center stack for cryogenic temps is nowhere near reality.

That said, once you have cryogenic superconducting integrated circuits you could colocate your data centers and your propellant/oxidizer depots. Not exactly "data centers off in deep space" since propoxd tend to be the highest traffic areas.