I found Pioneer Venus 2; December 9, 1978. Here we have an inversion of expectation; the primary mission was a set of hands-on Venus atmospheric studies. The bus had solar power but the probes in this case where the primary payload and were dropped into the atmosphere on battery power.
Pioneer Venus 1 (already present in orbit) was used as the relay.
Since I haven't found a future atmospheric probe (not intended for soft landing) launched on its own power but rather riding tag-along on a larger probe; I don't expect to find any newer answers. It is interesting to note that this was only a few years before the space shuttle that carried Galileo was launched.
NASA summary can be found in the expected place, but I'm mainly relying on the images from the Wikipedia article.
Excerpts from NASA mission summary:
Pioneer Venus 2, the sistership to Pioneer Venus 1, consisted of the main spacecraft, a Large Probe (698 pounds or 316.5 kilograms), and three identical Small Probes, all of which were designed to collect data during entry into the atmosphere of Venus. The probes were shaped like cones and were not expected to survive impact with the surface.Pioneer Venus 2 released the Large Probe (5 feet or 1.5 meters in diameter) on Nov. 16, 1978, while about 7 million miles (11.1 million kilometers) from the planet.
Four days later, the bus released the three Small Probes—the North Probe, Day Probe and the Night Probe—while about 6 million miles (9.3 million kilometers) from Venus.
Each probe took about 53 to 56 minutes to reach the surface. Amazingly, two of the three Small Probes survived the hard impact. The so-called Day Probe transmitted data from the surface for 67 minutes, 37 seconds, before succumbing to the high temperatures, pressures and power depletion.
The main spacecraft, meanwhile, burned up in the atmosphere at an altitude of about 75 miles (120 kilometers)—about 1.5 hours after the probes—and provided key data on higher regions.
Thus the probes, not the main spacecraft, are the primary payload; and thus they used batteries to achieve their primary mission. The use of battery power supply is obvious because you cannot use solar powers in the atmosphere of Venus and again because the images show no solar panels on the components expected to survive entry into Venus's atmosphere.