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@Dirk It is shown in the article by Elton and Odell which Bill Johnson has just mentioned (available e.g. from eudml.org/doc/266720) that $\beta(B_X)$ is always strictly greater than 1.
@Dirk I'm not actually sure if such spaces exist. I've now added examples that show $\beta(B_X) = 2$ also for $\ell^1$ and $L^1[0,1]$. Maybe $\sqrt 2$ is actually the minimal value for $\beta(B_X)$.
While the general Bishop–Phelps theorem is false for complex Banach spaces, it is worth noting that it holds for complex Banach space X with the Radon–Nikodym property.