FamilySearch has this record of her christening, though it does not agree with the researcher’s notes direct from the register:

name: Anne Emma Barrington
gender: Female
baptism/christening date: 15 Oct 1798
baptism/christening place: Calbourne, Hampshire, England
birth date: 12 Oct 1798
father's name: Fitz William Barrington
mother's name: Edith Mary
indexing project (batch) number: C01363-1
system origin: England-EASy
source film number: 1470803
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While she is in the baptism register, she does not appear again, certainly not as an heir of her uncle, Sir John Barrington, 9th bart, so must have died while an infant.

But see Note for Jane Elizabeth Barrington.

Apparently she was buried in Bristol cathedral: I am awaiting confirmation of this.

At All Saints church, Calbourne, Isle of Wight, transcribed by the IoW Family History Society and put on their website at <http://www.isle-of-wight-fhs.co.uk/mis_bis/mis_calbourne_all_saints.htm>;:

Anne Emma
Their Fourth Daughter
who died unmarried
the VIIth day of April MDCCCXIX
aged XX years
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