SSIS File System Task Copy File Could not find part of the path error

Mapped drives, like P:\ in your case are dependent on the user logged in. Just because you can see the drive letter when logged into the server machine does not mean SQL Server can "see" the drive letter. If you specify the UNC (Universal Naming Convention) name in the path instead of using the drive letter, and the account SQL Server is using has access to the share and path, then it will work.

Instead of P:\mypath\myfile.txt use \\machine\share\mypath\myfile.txt

if you did what Max Vernon have suggested, make sure to use "Rename File" operation in your script task (yeah, you will be moving the file)