I cannot find anything about her other then this portrait.
From the Christie’s Web site. Also posted to the Reign of Louis XV - 1715 to 1774 Album.
Lady Anne’s mildly inflated coiffure dates this to the early or middle 60s.
Keywords: Gainsborough, Anne Greenly, Anne Furye, straight coiffure, floral headdress, earrings, choker ribbon, choker neck band, sacque back bodice, bodice, lace modesty piece, lace-edged robings, close upper sleeves, bows, cuffs, stomacher, bows en échelle, lace engageantes