Basque is spoken by around three quarters of a million people in the western Pyrenees, on both sides of the border between Spain and France, and it is related to nothing. Every other language in western Europe belongs to a family that arrived: Indo-European overwhelmingly, with Uralic and Turkic at the margins and Semitic through Arabic and Maltese.
Basque was there first and stayed. Its immediate ancestor, or a very close relative, is Aquitanian, attested in personal and divine names on Roman-era inscriptions in southwestern Gaul, which shows the language in place two thousand years ago and no further back. Attempts to link it to something else have been made continuously for centuries. The Caucasian languages, Iberian, Berber and a hypothesised Vasconic substrate underlying place names across Europe have all been proposed.
R. L. Trask, who assessed the proposals in detail, found none of them sound, and the mainstream position is not that the question is contested but that there is no answer. The atlas records that as unknown rather than debated, and carries an open question saying what would change it. The language itself is nothing like its neighbours. It is ergative, it is heavily agglutinative, its verb agrees with more than one argument at a time, and its numeral system is vigesimal, counting in twenties. None of that proves isolation, since typology is not descent, and all of it is why the question keeps being asked.