You could glue(hide glue, carpenters glue or super glue) the strip back down and add a wrap at the beginning of the splinter. Sinew would be the best wrap material but silk thread or other thin, strong thread will also work. After you wrap, soak the wrap with super glue or other glue and let it all cure for a few days. Your bow will probably be a little stiff in the area where it is wrapped.
Pictures would help with the diagnosis.