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Additional evidence:
From https://www.wikitree.com/wiki/Tufton-48
In 1539, the Inquisition Post Mortem of Nicholas Tufton stated that he had a daughter named Alice whose surname at that time was Sharpe. He also had a daughter Agnes whose surname was Denne. [1]
Alice died in August 1568 and was buried on 31 August at the parish church of Northiam, Sussex: [2] [Anno Dni 1568 August] Alice Sharp vidua sepulta erat ultimo die Sextilis 1568. Alice Sharp Widow was buried on the last day of Sextilis [August] 1568
1. Nicholas Tufton in Notes of post mortem inquisitions taken in Sussex. 1 Henry VII, to 1649 and after. Abstracted and translated by F.W.T. Attree. Sussex Record Society Pub. 1912 Page 228. Accesssed 5 January 2020 Internet Archive.
2. "England, Sussex, Parish Registers, 1538-1910," database, FamilySearch (https://familysearch.org/ark:/61903/3:1:S3HT-DWL3-29W?cc=1465706 : 9 August 2016), > image 1 of 1; The Keep, East Sussex Record Office; West Sussex County Record Office, Chichester.
Apologies for delay in getting the surname updated.