The statement for complex scalars is true.  In the mid-to-late 60's, not long after Lindenstrauss' memoir was published, the theory of the "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 $X$ is isometrically 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 $X$ is a $P_{1+\epsilon}$ space for every $\epsilon > 0$ then $X$ has no proper essential extension.  

We use the criterion for essential extensions given by Lacey, p. 89: if $X\subset Y$ then $Y$ is an essential extension of $X$ if and only if the only seminorm on $Y$ which is dominated by the norm on $Y$ and equal to the norm on $X$ is the norm on $Y$ itself.

For a $P_{1+\epsilon}$ subspace $X\subset Y$, we define a seminorm $\rho$ on $Y$ by $$\rho(y) = \inf\{\|P(y)\|: P \: \text{is a projection of}\:  Y \: \text{onto}\: X\}.$$

The proof of the triangle inequality $\rho(y_1 + y_2)\le \rho(y_1) + \rho(y_2)$ uses the following lemma:  Given linear independent $y_1,\: y_2 \in Y$ and projections $P_1,\: P_2$ from $Y$ onto $X$, there exists a projection $P:Y \twoheadrightarrow X$ with $P(y_i) = P_i(y_i),\: i=1,2$. (Here we use the fact that $X$ has the extension property: bounded linear operators into $X$ can be extended.)  Since $X$ is $P_{1+\epsilon}$ for all $\epsilon > 0$, we have $\rho(y) \le \|y\|$ on $Y$ and $\rho(x) = \|x\|$ for $x\in X$.  But if $Y$ is a proper extension of $X$, $\rho$ can not be equal to the norm on $Y$.  Thus $Y$ is not an essential extension by the criterion mentioned. QED

Notice that this proof is valid for real and complex scalars.