It is well known that a real Banach space which is $(1+\epsilon)$-injective for every $\epsilon >0$ is already 1-injective (Lindenstrauss Memoirs AMS, 1964). Using common terminology, If $E$ is a $P_{1+\epsilon}$-space for every $\epsilon >0$ then $E$ is a $P_1$-space.

The proof of Lindenstrauss seems valid only for real scalars. Has a proof of the corresponding statement for complex scalars appeared in the literature?

This result is easier if $E$ is a dual space, and the proof in Semadeni's book seems to work for complex scalars.