But, here is the earliest school I can find, Ellinthorpe (see Marjorie R. Theobald's book, Knowing Women). The school was established in 1823 by a most intrepid woman, Hannah Maria Davice, who emigrated with a friend, Elinor Binfield, and a younger relative, Susannah Darke Purbrick in June, 1823. Hannah opened her school in rented buildings in Hobart Town within a month of arriving. She married George Carr Clark in December 1824 and continued to teach, changing locations to Carr Field House opposite the Post Office in Hobart, and then to Ellinthorpe in September 1827, described as a rural Victorian estate. Apparently, Lady Jane Franklin, wife of the Lieutenant Governor of Tasmania, was not too supportive, having educational projects of her own, but the Australian Dictionary of Biography reports Ellinthorpe was the most prestigious school in the colony, serving about 40 girls.
Hannah closed her school in 1840, taking her 6 children back to England for further education and leaving her husband behind. But Susannah Purbrick (Mrs. John Knight) opened her own school, Carr Villa, near Launceton in 1848 until 1866. I'm on the hunt for more details about how these schools worked and who the students were. That gap in 1842 might very well be useful for my story. And back to work it is, with more than three clicks!
Source of picture of Ellenthorpe Hall.