In 1842 neither she nor any children of hers were listed as heirs in a document on some Barrington property, probably because she was already dead and had no children.

But see Note for Jane Elizabeth Barrington.

Apparently she was buried in Bristol cathedral: I am awaiting confirmation of this.
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Family Search has this record of her marriage:

groom's name: Thomas Pakenham Vandelear
bride's name: Mary Barrington
marriage date: 05 Aug 1827
marriage place: Calbourne, Hampshire, England
indexing project (batch) number: M16647-1
system origin: England-EASy
source film number: 1470803
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This notice was printed in an Irish paper on 11 June 1929:

“At Clifton, in the 21st year of her age, Mary, the wife of Captain Thomas Pakenham Vandeleur (late of the 21st Fusileers) and daughter of Sir Fitzwilliam Barrington, Bart. of Swainston, Isle of Wight.”
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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>;:

Mary
Their Sixth Daughter
and Wife of
Thomas Pakenham Vandelour Esqr.
of Limerick in the Kingm of Ireland
who departed this life
the XXIX day of May MDCCCXXIX
Aged XXI years.
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Tis was on a Sunday.