A’Campo showed that the finite symplectic group $Sp(2m,p)$, $p>2$ prime, is a quotient of the braid group $B_n$ for some $m$ depending on $n$. Hence the finite groups $PSp(2m,p)$ are quotients of $B_n$. These groups are simple non-abelian hence perfect for most $m, p$.
A’Campo, Norbert, Tresses, monodromie et le groupe symplectique, Comment. Math. Helv. 54, 318-327 (1979). ZBL0441.32004 MR0535062