She belonged to the Ásbirningar family clan and was the daughter of Arnór Ásbjarnarson and sister of Kolbeinn Arnórsson.
She was the first woman in Iceland to receive a formal academic education and to serve as a teacher.
Ingunn Arnórsdóttir was a student at the Latin school of Bishop Jón Ögmundsson (reign 1106–1121) at Hólar.
She was the only female student at the school and the first woman in Iceland to study Latin and have an academic education.
Ingunn Arnórsdóttir was also one of the sources for Ólafs saga Tryggvasonar, a biography of the Norwegian King Olaf Tryggvason, written by the 12th century monk Oddur Snorrason at Þingeyrar.