I had one go under my thumb nail, didn't realise it was there but my thumb went infectious.
Doctor said she had to pull my nail off or I could do it, so I stuck a hot skewer in my nail and lifted it off, it was the only way to remove it.
Took a year to grow back and was very sensitive all that time. Don't recommend leaving a carbon splinter inside at all, nothing like a metal or wooden splinter.