Imagine a liquid which can be electrically charged, and has a low boiling point.
(Ask 3M/DuPont/BASF/Bayer... - context 'immersion cooling')
Attach heat-pipes with that stuff to the chips as is common now, or go the direct route via substrate-embedded microfluidics, as is thought of at the moment.
Radiate the shit out of it by spraying it into the vacuum, dispersing into the finest mist with highest possible surface, funnel the frozen mist back in after some distance, by electrostatic and/or electromagnetic means. Repeat. Flow as you go.
Even if that worked, you don’t gain much. It’s not the local surface area that matters — it’s the global surface. A device confined within a 20m radius sphere can radiate no more heat than a plain black sphere of the same radius.
There are only two ways to cheat this. First, you can run hotter. But a heat pump needs power, and you need to get that power from somewhere, and you need to dissipate that power too. But you can at least run your chips as hot as they will tolerate. Second is things like lasers or radio transmitters, but those are producing non-thermal output, which is actually worse at cooling.
At the end of the day, you have only two variables to play with: the effective radiating surface temperature and the temperature of the blackbody radiation you emit.