The answer is **no** and $E=\mathscr{K}(\ell_p)$ for $p\in (1,\infty)\setminus \{2\}$ is already a counterexample. See the proof of Proposition 2.3 in

> J. Hennefeld, [Compact extremal operators][1], *Illinois J. Math.* **21** (1997) 61-65.

Here we identifty $\mathscr{K}(\ell_p)^{**}$ with $\mathscr{B}(\ell_p)$.


  [1]: http://projecteuclid.org/euclid.ijm/1256049500