Assuming (like Retudin's answer) that smallest P means smallest polyomino, and that placements of polyominoes have to be perfectly grid-aligned:
The optimal solution is P=2, with dominoes.
The reason why a domino solution is optimal is that a finite solution with grid-aligned monominoes is impossible (thanks to @AxiomaticSystem for giving me the idea). Note that if the solution is allowed to be infinite, there are a lot of trivial solutions.
Why? Consider that for every monomino solution, there must be a top row of the solution (that contains at least one monomino). Then we consider one of the monominoes in the top row. Since it must be adjacent to three other monominoes and there cannot be any monominoes above it, both the left and right cells must have a monomino.
Both of these cells are still on the top row, so we can repeat the same argument ad infinitum, getting that every cell in the top row must have a monomino in it. However, this contradicts the premise of the solution being finite. Therefore, a monomino solution is impossible.