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Question: What spacecraft relied on solar electrical power farthest from the Sun?
If you like you can break it down by use case; spacecraft that used it for solar-electric propulsion (and weren't just coasting while hibernating) versus those that had conventional propulsion but still used solar power for electricity.
According to Wikipedia the Juno mission to Jupiter is "the farthest solar-powered trip in the history of space exploration". It has three solar panels, each of which measures 2.7 by 8.9 meters.
Juno uses the electricity generated by the solar panels to power its heaters, instruments, on-board computers and communications. It can store excess power in two $55$ amp hour lithium ion batteries. Its propulsion system is a conventional combination of main thruster and $12$ small attitude control thrusters powered by hydrazine propellant.
See also How Juno Broke The Distance Record For Solar-Powered Spacecraft
It's Rosetta or Juno:
In June 2011, Rosetta was placed in hibernation as it made its way beyond the orbit of Jupiter where there was no solar energy to power the vehicle.
Despite the wording of the above statement, the solar panels did continue to operate during hibernation, and supplied power to the spacecraft's computer and heaters.
Rosetta reached an aphelion of 792 million kilometres (5.29 AU) from the Sun on 3 October 2012. This is outside Jupiter's perihelion, but not its aphelion. Jupiter is currently 783.6 Mkm from the Sun, so the record depends on where in Jupiter's orbit we've been since Juno's insertion.
As for solar electric, that would be Dawn, which used it to explore the asteroid belt.
Juno is certainly the furthest with chemical propulsion, the only mission to be anywhere near Jupiter and use solar power.
Supplemental answer necessitated by the question of exactly how far Rosetta might have used its ion engines from the Sun found in comments below both of the other answers.
edit: I just found that these distances can also be confirmed in this answer.
I downloaded positions for Rosetta, the Sun, Earth, and comet 67P from JPL's Horizons at 1 day intervals and plotted them below.
Using the dates found in this answer for the beginning and end of hibernation 2011-06-08 and 2014-01-20 corresponding to distances from the Sun of 4.46 and 4.49 AU, there's no question that Juno at Jupiter with it's aphelion of 5.46 AU is unquestionably the winner.
Python plotting script: https://pastebin.com/wqrL3aQr
