No. Under the REAL ID Act, one must have a valid, unexpired nonimmigrant visa.
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In Florida you must prove that you are lawfully present in the U.S. This means being in valid immigration status. Since you overstayed you are not in valid status and therefore ineligible for a state identification card or driver's license.
While this answer is provided by a Florida Bar Certified Expert in Immigration and Nationality Law, it is for general information purposes only and an attorney/client relationship is neither intended nor created. You should seek out qualified counsel to review your case and provide you with advice specific to your situation. Call +1-561-478-5353 to schedule a consultation with Mr. Devore.