Yes, I am confident that all materials get warmer when exposed to sunlight. Evaporative cooling might be argued as an exception, however I do not consider it so.
There might be some materials that do not absorb solar radiation, I'm not aware of them.
There is radiative cooling paint that in addition to reflecting most light radiates heat in wavelengths that don't get absorbed by the atmosphere. That makes them cooler than ambient temperature when they are under the open sky.
They still get heated by the sun, just less so than everything else. But very similar sounding sentences about them would be true (because on a sunny day, in the shadow, they would be cooler than indoors at the same ambient temperature)
I'm pretty sure the cheapest mirrors still manage 80% with good quality ones in the 95-99% and 99.999% achievable if you're prepared to get fancy. So I'd say they're pretty dang close to perfect. Definitely well down the asymptote.
I don't get what point you're trying to make... They reflect some radiation depending on quality. Regardless how close to perfect they get, the answer to the original question is - no, they don't break the "everything heats up" rule.
There might be some materials that do not absorb solar radiation, I'm not aware of them.
https://en.wikipedia.org/wiki/Evaporative_cooler