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 $\mathscr{B}(\ell_p)$.

This question is discussed in:

> S. Dutta, T. S. S. R. K. Rao, [On Weak-${}^\ast$ Extreme Points in Banach Spaces][2], *Journal of Convex Analysis*, **10** (2003), No. 2, 531-539.

where further examples are given.


  [1]: http://projecteuclid.org/euclid.ijm/1256049500
  [2]: http://home.iitk.ac.in/~sudipta/revise.pdf