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 threadWe 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.
(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.