[J. Lindenstrauss and L. Tzafriri. Classical Banach spaces I. Sequence spaces. Springer 1977]. On page 76, after prop. 2.c.3, it is stated that the proof of 2.c.3 shows that an operator $ T: ell_p to ell_p $ is strictly singular if and only if it is compact.
[F. Albiac and N. Kalton. Topics in Banach space theory. Springer 2006] Theorem 5.5.1 says that a weakly compact operator $ T: C (K) to X $ is strictly singular, and Theorem 5.2.3 says that a non-weakly compact operator $ T: C (K) to X $ it's not strictly singular.
Note that $ ell_ infty $ is a $ C (K) $ space with K $ Stone-Cech compactification of all positive integers.