The Invisible Man - H. G. Wells, 1897
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In discussing the process used to turn the protagonist invisible, Wells writes:

> ...the essential phase was to place the transparent object whose refractive index was to be lowered between two radiating centres of a sort of ethereal vibration, of which I will tell you more fully later. No, not those Röntgen vibrations—I don't know that these others of mine have been described. Yet they are obvious enough.

"Ethereal vibrations" is a 19th century term meaning radiation. (This is from a time before Einstein had pinned down the interpretation of the [Michelson-Morley experiment](https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment), and radiation was thought to be due to vibrations in an ever-present "ether".) "Röntgen vibrations" refers specifically to X-rays, which had been discovered only two years previously. 

So Wells' protagonist got his invisibility "super power" by exposure to some form of radiation similar to X-rays, but different. It's a striking early example of unusual abilities being associated with exposure to radiation.

There is one little technical detail that might make this debatable: we don't actually know if the radiation in Wells' story was ionising or not. This question isn't addressed in the story, and couldn't be, because such a concept didn't quite exist in science at the time. (The electron was discovered in the same year; gamma rays wouldn't be discovered until 1900.) But given that you only specified "ionising radiation" in order to avoid mundane things like sunlight, I suspect this counts as the earliest example of what you're looking for.