Tjunkiya was born around 1927: the main biographical reference work for the region gives a date of circa 1927; while the Art Gallery of New South Wales suggests circa 1930.
A Pintupi speaker, Tjunkiya was born in the area northwest of Walungurru (known as Kintore, Northern Territory), near the Western Australian border, and west of Alice Springs), after which her family moved to Haasts Bluff. She became second wife to Toba Tjakamarra, father of one of the prominent founders of the Papunya Tula art movement, Turkey Tolson Tjupurrula. At Haasts Bluff she had ten children: these included sons Billy Rowe and Riley Rowe, both of whom painted for Papunya Tula, and daughter Mitjili (born c. 1948), who married Long Tom Tjapanangka and went on to paint at Haasts Bluff. From Haasts Bluff the family moved to Papunya and in 1981 to Kintore.
Tjunkiya was the sister of artist Wintjiya Napaltjarri, who was also a wife to Toba. Seriously ill in the mid-1990s, Tjunkiya died in 2009.