A new language for the Dark Ages, Basque-Icelandic came into being about the year 775. The Moorish conquest of Iberia had unsettled the Visigothic Kingdom, leading to a northwards diaspora. As with the differently caused but mainy analagous Irish Missionaries, Basque traders spread basque culture throughout the coasts of western Europe. However, they did not leave a lasting influence. Carolingian power in France, coming from the line of Charles Martel, had left Gaul unfriendly to the newcomers. English Barbarians likewise did not do any largescale trading.
However, Iceland was more hospitable. Both the Icelanders and the Basques were outside the mainstream of Western life, and traders ploamed the western Atlantic. Naturally, Basque-Icelandic pidgeon developed.
Though unproven, an interesting theory is the pidgin is responsible for the poor translation quality of ancient Greek and Arabic texts found on Iceland (obviously, the colanguage of such distinct groups would lead to many losses in the translation].
No written scraps of BIP survive after 812, and no reference to it is made as a contemporary language after 936 (though in all probabily it had probably gone extinct long before).