Yuichiro's is the best answer, but here's another "cheat" answer (which I nonetheless think is kind of neat):
Take a PBD$(v,\{3,5\})$, triple each block of size three, and replace each block of size five by the complete design on 5 points$\binom{[5]}{3}$. Various PBD$(v,\{3,5\})$ can be found by cyclotomy.