The short answer is NO.

Explanation: Unlike in most other countries, where you'd need both a driver's license AND an identification document, in the US, your driver's license is your ID. If you don't have a driver's license, you can get an ID issued to you by the FL DMV or Tax Collector's Office. What happens if you have FL Driver's license, and you request a FL ID? You will be issued a Florida ID, but your Driver's License will be cancelled. There are three types of Florida Identification documents, and you can have only one at any given time. These are:
Florida Identification Card
Florida Learners Permit
Florida Driver's License
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