Savate seems to have its origin as a collection of informal, uncodified streetfighting tricks, apparently emerging in the late 1700s and undergoing a very gradual process of codification for the best part of the next fifty years, before it was finally systematised and presented to the dominant (literate, middle-class) culture, by Michel Cassaux (1794-1869).
It definitely does seem to have combined elements of sport and street self defense from the earliest times, with the street fighting aspects (la defense dans la rue) reaching their highest point of development in the late 1800s and early 1900s, while the sport aspect (la boxe Francaise) has continued to develop to the present day.
Basically there are three competing theories about the origins of savate;
1 - that it's a survival of ancient Mediterranean/European combatives (q.v. Pankration, zipota, purring (British/Welsh shin-kicking sport), etc.);
2 - that it was influenced by Asian martial arts, presumably those encountered by French sailors travelling in the Far East;
3 - that it was influenced by African and African-derived martial arts/combat sports witnessed by French travellers, either on the African mainland or in French-controlled outposts such as Martinique
The truth is probably "all of the above", but AFAIK no-one knows for sure. Those few academically-oriented researchers who have taken an interest in the art still argue about its ultimate origins, and most of that discussion has been in French, meaning that even after three hundred years, comparatively little solid information about the origins of Savate is available to the English-speaking public.