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The statement for complex scalars is true. Shortly after Lindenstrauss' memoir was published, the theory of injective hull of a Banach space was completely worked out. It is pretty well covered in Section 11 of Lacey, "The Isometric Theory of Classical Banach Spaces". There we see that every Banach space $E$ is embedded with an "essential embedding" into a unique $C(K)$ space where $K$ is extremally disconnected (Sometimes called "Stonean"). For our theorem, it therefore suffices to prove that if $E$ is a $P_{1+\epsilon}$ space for every $\epsilon > 0$ then $E$ has no proper essential extension.