Ah, very interesting, thank you! Is it possible that they traveled to Provence during their holidays or to visit relatives/friends and picked it up there? My guess is that they were quite affluent and I think going on a holiday to another region wouldn't have been unusual. Then again I don't know very much about the time period.
I also find this, which shows usage since 1610:
https://www.geneanet.org/prenom/Anais