I didn't manage to stay upright, but I managed to slow down, so I only went down when I hit the corner, and I slid down the wood. Wood burns on my arm, nothing else, fortunately.
I am using brand new wheels with carbon rims built by a professional. What I blame it to is the latex tube I was using. I inspected rim and rim tape and they look perfect. It wasn't a splinter, neither I think it was the rim tape, most likely it was the latex tube being slightly pinched between the rim and the tyre bead. For now, I am going to switch to butyl tubes, I don't care about those 5 extra watts anymore, I care more for safety. But I am also no longer confident about clinchers on the track in general. I have never used tubulars, but as far as I know, they can't blow up so quickly.