Can I apply for a Canadian study permit after I have submitted the PR?
Logically speaking, I don't think it will be a good idea. No one is stopping you to do that though. There's no law or regulations as such. But that will hinder your chances for PR. If you are unsuccessful in achieving the PR. Than you can take the other route as a student. But for now I would suggest to just have patience and wait for the results of your initial application. If you succeed in achieving your PR, you can always come and study and you won't have to pay the hefty tution fees. Tution fees for international students are almost double.