They are considered a subspecies of the White tail deer, and they are believed to be small because of stunting due limited resources on the islands. So yes, they are classified as the same species. However they have been in the keys and small in size since long before people were here so to call them an invasive species is simply incorrect and so would eradication, as they do play a vital role in the local ecosystem. (more info here for anyone interested)
They do tend to spark a lot of debate among people though, an interesting study has been done about this conflict.