Sticky Note Square off!

Daniel and Cody are playing a game on an infinite unit square grid. Daniel places unit square sticky notes with a letter $D$, while Cody places unit square sticky notes with a letter $C$. They take turns placing sticky notes on squares in the grid, with Daniel going first. Cody wants to have four of his sticky notes form the corners of a perfect square with sides parallel to the grid lines, while Daniel wants to prevent him from doing so.

Can Daniel succeed?

Assume that both players make optimal moves.