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, Illinois J. Math. 21 (1997) 61-65.

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

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, 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, Journal of Convex Analysis, 10 (2003), No. 2, 531-539.

where further examples are given.