Is there another prime $p$ such that $S(p)$ is prime?

Question

Denote $$S(p):=2^2+3^3+5^5+\cdots +p^p$$

$S(p)$ is prime for $p=3,7,89$.

Is there another prime $p$ such that $S(p)$ is prime ? Is the number of primes $p$ such that $S(p)$ is prime, finite ?