Is plain multiplication a P problem?
Yes, and with many different algorithms. 

See https://en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations

One important thing to notice is that the size of the input is the number of digits in the numbers, rather than the magnitude of the numbers (e.g. the "size" of the number 657 is 3, rather than 657, because it has 3 digits).