There are a few examples of Alixes after the Middle Ages and before the name was revived by Princess Alice:
The Blessed Alix Le Clerc (1576 - 1622), a Lorrainian nun who founded the (Chanoinesses de Saint-Augustin de la) Congrégation Notre-Dame dedicated to the education of girls.
'Madame Alix', aka Léonne-Julie Bournonville (1748 - 1826), famous ballet dancer who also performed at the Mariinsky, aunt of Auguste Bournonville, married to Claude Alix de La Faye, dentist to the Queen of sweden.
(Françoise-)Alix de Lamartine (1776 - 1829), writer and mother of the famous poet Alphonse de Lamartine.
Alix Joffroy (1844 - 1908), French psychiatrist, NB a man. (Alix is also both a placename and a (rare) man's name in France.)