Transcoding from m4p to wav to mp3 is a direction a lot of people have taken simply for the sake of portability between different environments. But as Christina says, converting an m4p file to another lossy format will result in some loss of audio information. Depending on your settings and the other factors I mentionned previously, this loss may be imperceptable. And the practical advantages in having a more portable file that you can use in more ways may well make the m4p to mp3 conversion a good option despite some loss of audio quality. If disc space is no object then converting m4p to wav to other lossless codec will preserve the quality of the original m4p the best but these files will be larger than the original m4p file.
One point to remember is that all other considerations being equal, any audio file (lossy or lossless) will always be its own best copy.
Another is that a difference that does not make a difference is not a difference. Even if there is a loss of audio quality in converting from mp3 to mp4 or the other way around, if you cannot tell a difference then it maybe it shouldn't be a worry. I am reminded of a song by the duo of Flanders and Swann called "High Fidelity", which dealt with the fascination of the then-new stereo sound. One verse in particular is relevent here:
"High Fid-el-ity,
Hi-Fi's the thing for me.
All the highest notes
neither sharp nor flat.
The ear can't hear
as high as that.
Still I ought to please
any passing bat
with my high fidelity!"
Best wishes,
Bill Mikkelsen