So far, I have failed to find any other information about Elizabeth. The lack of information available seems to suggest that she may have died in infancy.
1 GS Film Number: 918429. Digital Folder Number: 007588075. Indexing Project (Batch) Number: C10557-2. System Origin: ODM. “England Births and Christenings, 1538-1975”, database, FamilySearch (https://familysearch.org/ark:/61903/1:1:J796-D6V : 19 September 2020), Elizabeth Norton, 1808.