It is difficult to know which girls' school was definitely the first in Peterborough. The first for boys, The King's School, was founded in the reign of Henry VIII, but girls were not deemed to need educating, unless they were wealthy. They were educated in skills that were seen to make them more attractive and have more chance of marrying.
In the Georgian period Dame schools started to appear. These were schools run by women to teach girls useful skills like sewing and dancing, as well as reading, writing and simple maths. The girls usually boarded with the women running the schools in large houses. Unlike modern boarding schools the number of girls would have been relatively small and dependant on the size of the building.
Mrs Elizabeth Bacon was the headmistress of the first girls boarding (Dame) school in Peterborough. The first record of the school is from 1753. She ran the school until 1770 after which when Miss Searle took over as head teacher. (1)
Other Dame schools included Mrs Horden's (see other entry) and a girls school run by Miss Mary Smith in 1791.
(1) D.K. Shearing, Education in the Peterborough Diocese Following the 'Glorious Revolution' 1688, (unpublished PhD Thesis, University of London) p289 via http://discovery.ucl.ac.uk/10018490/1/121273.pdf