No. When a cell splits, it BECOMES 2 'new' cells. The original cell no longer exists. The daughter cells can split and make new cells. This can happen for millions and millions of cycles.
Technically, a cell can create infinitely many daughter cells. However, and I'll use the human body as an example--if your cells had uncontrolled division like that, you'd have cancer. Cells that work together, like those in the body, only divide when they need to (i.e., during growth or to fix a wound).