We see this paradox being "resolved" in Zee and P&S through a careful (if still pretty confusing) translation from a field propagator to a real, measurable "particle". For instance, Zee claims that the leaking of a field outside the lightcone does not imply a "real" particle being detected there. Also, I can't recall if some part of the apparent paradox came from adding over momenta which are far from the mass shell.
Peskin and Schroeder demonstrate that free quantum fields (anti)commute at space-like separations. However, there is no relationship between this fact and the speed of propagation of wavefunctions. In a very long thread
(starting somewhere in the middle of the thread) there is a discussion of the relationship between wavefunctions and quantum fields. I still believe that these are completely unrelated things. In other words, I don't take what's written in P&S as a convincing proof.
I don't have Zee's book, but I doubt that their explanation is fundamenatally different from that in P&S.