We assume all m4p files are protected. That is what the "p" in m4p stand for, isn't it? If a file is not protected, its extension should really be m4a, not m4p.
This is correct. There is no difference between MP4, M4A, M4V, M4B, M4R, M4B, and M4P. All of them except for MP4 were file extensions that Apple invented to differentiate between the different uses of MP4 files. They are not different kinds of files. They are all just MP4s with different kinds of content inside.
MP4 is the "real" standards-body approved file extension.
M4P means simply a "Protected" (DRM-laden) MP4.
M4V is a MP4 that contains video content.
M4A is a MP4 that contains audio content.
M4B is an audiobook (usually also DRM-laden).
M4R is a ringtone.
So on and so forth. There is absolutely no need to "convert" between a M4P and a M4V. If the file really does not have DRM, you can just change the file extension to either M4V or MP4. They are the same exact thing.
Now, I'm really not sure where you got a M4P file that has no DRM. That's weird.