The first part of the word viking, or vik simply means "bay" in nordic languages

Do we have any historical source of people referring to themselves as vikings?

I don’t speak old norse but I speak Icelandic natively. Víkingur simply means Bay-er, that is somebody from a bay. As an Icelander living in America I experience the English word “viking” as an Exonym for my identity. In Iceland we use “Nordic” or “Scandinavian”, both terms are inclusive of Finns, Sámi, and Greenlanders, so strictly speaking this is not an Enthnonym.

In Icelandic, at least to my knowledge, we have never used Víkingur as an ethnonym (well maybe during a sports game, or among right-wing nationalists). It has always meant raiders. In 2007 there was even a new word dubbed Útrásarvíkingar meaning businessmen who made a bunch of money doing business abroad (buykings would a clever translation of the term).

EDIT: I just remembered that the -ingur suffix can also be used to indicate a temporary state e.g. ruglingur (confusion) and troðningur (trampling [n.]), and was used as such e.g. að fara í víking (to embark to a viking) so víkingur could also mean, a person that embarks to a bay.