Yes, and the space you gain may cost you much more than simply adding the same amount of space to your house above ground.
The basement must be hand-dug another three feet, and the soil caried out pail by pail...VERY labor intensive.
Then, as you dig, the existing foundation nmust be underpinned. This involves digging below the existing footings in 3-foot widths spaced 6 feet apart, placing forms, and filling the forms and excavations with concrete, then digging the next 3 feet, and repeating the process, and finally digging the last 3 feet and doing that. Also VERY labor íntensive.
Then, if you plan do develop habitable space in the basement, there must be a second means of egress from it, besides the stairs to the main floor. And then the basement must be insulated.
Finally, habitable space in the basement must meet code requirements for natural lught and ventilation.
Those are the cons.
The pro is that if you have absolutely no way to add to the house above ground, and you absolutely need the space, and you absolutely can't or don't want to buy a bigger house, then it becomes a feasible alternative. It just won't be inexpensive.
Architect (NY) and Home Designer (PA)