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, download here). 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.

[Edit 7/1/2013] After 1 month and 138 views, no answer is posted. Two experts (not on MO) have told me they did not know of a reference. This is likely not in the literature, which is somewhat surprising to me.

With editorial license, I am changing the question: Give a proof of the statement for complex scalars.

[Edit 8/20/2013] Per a reader's suggestion, be warned that the proof I offered below turned out to be incorrect, as mentioned at the end. Maybe the idea can be rescued.