After a close look of the stem it may be a replacement stem. The "p" looks very much like what I have seen done on replacements. Also I could be wrong on this one, but I dont think Peterson ever stamped the "p" on the system pipes?
Being a pipe restorer from Ireland, I have handled literally hundreds of these pipes from all eras. These are my thoughts.
As quoted above, I have NEVER seen a P logo stamped on a system stem. Also, that stem doesn't seem to be seated correctly in the shank. I bet the bowl would fall off the stem easily. Replacement Peterson stems are not a one-fits-all solution. Every shank is different and it is best if replacement stems are custom fitted.
Also, as regards Petersons - the older the briar the better. The markings on the nickel collar points this out to be fairly modern production. The best briar is in the Pre-Republic pipes so always look out for these within your budget. If your budget doesn't stretch to a pre-Republic model, go for the ones that have the 3 faux hallmarks on the collars of the shamrock, tower and prone fox. These will give a much better smoke than the modern era pipes.