Genesis 16:12 has Ishmael described as 'a wild ass of a man' so I think that it would be unlikely that the Arabs would identify with the character. Is there any evidence of the ancient Arabs calling themselves Ishmaelites?
I am also skeptical of Islamic genealogies of the Arabs, because they were compiled retroactively in order to justify the legitimacy of Arabs as legitimate inheritors of the Abrahamic tradition. (In short, I think they are apologetic rather than historical.)
I would be grateful if anyone has any literature on this as well.
Is there any evidence of the ancient Arabs calling themselves Ishmaelites?
The term Ishmaelites appears mostly in non-Arabic sources. While a genealogical link between the Arabs and Ishmael appears in numerous Arabic sources, Arabs nonetheless did not use that term (the Arabic equivalent of the term, Ishmaelites, is often used to refer to a specific group within Shi'ism).
Genesis 16:12 has Ishmael described as 'a wild ass of a man' so I think that it would be unlikely that the Arabs would identify with the character.
We should not assume that any link between the Arabs and Ishmael was made to fit that description particularly. That's to say, when the Arabs were identified/self-identified as Ishmael's descendants, this identity was not necessarily made in light of the biblical view. This should especially be taken into account when approaching Arabic sources. For example, the Qur'anic account of Ishmael is very positive. Ishmael is portrayed as a patient, obedient and helpful son, an exemplar father-son relationship.
The earliest surviving source to link the Arabs with Ishmael is a second-century BCE work ascribed to Artapanus (of Alexandria). Speaking of the Arabs, he wrote:
The kings of the Arabs were the descendants of Ishmael, the son of Abraham and the brother of Isaac.
The first Arabic source is the Qur'an. A passage from Q 22:78 reads (translation by M. Abdel Haleem):
He has chosen you and placed no hardship in your religion, the faith of your forefather Abraham.
EDIT: I forgot to mention this: we should distinguish between two terms here, Arabs and Arabians. Arabs are Arabian, but Arabians are not necessarily Arab. The different groups of Arabia are called Arabian, among them one group is known as Arabs. A term used by the group itself. Arabic sources speak of the Arabs specifically as Ishmaelite, not the Arabian groups in general. For example, Ibn al-Kalbī (d. 819 CE) in his writings about genealogy links some Arabian groups to the biblical Joktan (Yaqṭān).