I might be misunderstanding this, but...I looks like you might be using the word 'radiation' in two separate ways?
The radiation at Chernobyl is 'ionizing radiation'. It has high enough energy to kick out electrons from around their atoms. Therefore, it can interfere with biological molecules (like genetic material), which is what makes it dangerous.
On the other hand, solar radiation is, for the most part, non-ionizing. It might be very /intense/ (as in, turning the sun-side face of a tidally locked planet into a molten desert), but for the most part individual photons don't have high enough energy / frequency to mess with chemistry. The exception to this are high-energy UV rays (you could get plenty of those on a tidally locked planet, if you want) and X-Rays (events called solar flares prduce X-Rays, but I think having an atmosphere blocks it, and presumably you'd want an atmosphere). So you might easily enough have organisms that shrug off high-energy UV rays, but they won't survive Chernobyl. It might vary a bit depending on the star, but for the most part that's it, I think.
One exception to this is pulsars. Pulsars are what's often left of giant blue stars after they explode. They're very small, very dense stars (imagine neutrons piled to the size of a mountain). They...uh...emit ionizing radiation like nobody's business. Like, for a subset of them (called magnetars), you can detect their magnetic field from /halfway across the galaxy/! At that point, you don't need your planet to be tidally locked. It's going to be blasted with ionizing radiation either way.
Whatever life develops there...would be very, very strange. There'd be so many things to consider, such as the amount of light you got (probably not much, might need to supplement the warming of the planet with hydrothermal vents)...which I guess you might get around by having your planet in a binary system, with one of the stars being a pulsar...and I'm wondering if that would work because sometimes pulsars /blow nearby stars to bits/...but...yeah. Lots to think about.
I'm wondering if another solution might be to have a planet in a late-generation solar system. From the little I know, stars from from clouds of dust and gas called nebulas, exist for a while, then some of them explode and this forms other nebulas, then time passes and more stars form. However, as these generations of stars occur, the elements involved (in the formation of the star and the planet) get heavier and heavier. I remember reading in a sci-fi book that planets in later star generations might have too many heavy metals (like uranium, which has useful radioactive isotopes!) to be comfortable for humans. Can't attest to the accuracy of that. Then again, you don't need your planet to be tidally locked. You'll have a planet that's largely radioactive. And...maybe it would make sense placing it in a more central part of the galaxy, where more star generations would've occurred...but that's just a guess and would have to be checked.