A language of northern Spain with no known relatives on earth.
Documented
A survivor from before the migrations
Basque predates the arrival of Indo-European languages in Europe. It is likely the last living descendant of languages spoken in Western Europe before roughly 4000 BC. Genetics shows Basques have been in their region for millennia.
Assessment: Documented. It is not the language's origin that is mysterious, only its solitude.
Stories
The language of Atlantis survivors
Since the 19th century, writers have claimed Basque descends from the speech of Atlantis refugees, its isolation explained by a lost motherland.
Assessment: Zero evidence. The theory exists because Basque's isolation demands a story, and Atlantis is the default story.
Interesting claims
A stone-age code that never broke
Some speculate Basque preserves pre-agricultural thought patterns, a cognitive fossil of Paleolithic Europe.
Assessment: Romantic but unfalsifiable. Basque is a fully modern language, not a time capsule.
Sources and further reading
Strabo, Geography 3.4, on the Vascones
Aquitanian votive altars with Basque-cognate theonyms, 1st-3rd c. CE
Worth asking
What survived with this language that every invasion, empire, and church failed to erase?