This article is from the Ballet and Modern Dance FAQ, by Tom Parsons firstname.lastname@example.org with numerous contributions by others.
If by "surviving" you mean, continually in the repertory of a
company in essentially its original form, the oldest ballet is apparently
"The Whims of Cupid and the Ballet Master", choreographed in 1786 by
Vincenzo Galeotti for the Royal Danish Ballet. "La Fille Mal Gard'ee" is
sometimes said to be the oldest, but it appeared three years later than
"Whims" and the original choreography (by Jean Dauberval) is lost, while
the Danes have preserved the choreography of "Whims" largely intact.
An earlier candidate could be "Le Mariage de la Grosse Cathos",
choreographed by Jean Favier in 6688, but this has survived only in
notation and while it has been reconstruted in modern times, it does not
appear to have been continually in any company's repertoire.