Excellent commenter mst3kharris brought up the point --
I'm taking it out of the comment trail and posting it here because the answer got long.
Hi mst --
Oh dear. I didn't mean to be confusing.
I used 'Anneka' as a place-holder name for my heroine. A 'working name'. It's the name that made her real for me and it's still how I think of her.
But I knew I wasn't going to get away with using this name in the actual book,
because Anneka is not French.
It's kind of funny. About the first thing my editor said to me when we were talking and getting to know each other was, "You know you're going to have to change Anneka's name, don't you?"
"I know," says I.
Now, all along, I saw the Anneka character as having a baptismal name and having a nickname that she actually used on a day-to-day basic. I wanted both a 'core', original identity for her AND a nom-de-spy.
I liked 'Anne' as the baptismal name because, in exactly that form, it would work for a Frenchwoman or a Brit. Many names have different French/English forms.
So I started from there. From Anne.
So they slid 'Annie' over to another common diminutive of Anne
-- Ahn EEK'
This name, under fifty different spellings, is common all across Europe. In Brittany, where they would have landed, it's spelled Annick. In France itself, Annique.
I used the French, (and more modern,) spelling of the name rather than the Celtic spelling, because I didn't want readers getting all distracted wondering why the heroine didn't have a French name, her being French and all.
Would the heroine have been known by a nickname?
I do think so. French baptismal names -- then and now -- are traditional and stuffy and generally involve being called after your great-aunt Eglantine-Claudette.
(Eglantine means 'needle'. Why would anyone name an innocent baby, 'needle'? Though I suppose it would be a good name for a junkie.)
But I digress.
So lots of folks in France operated under various sorts of 'use names', their names on the baptismal papers being truly awful, even by the elastic standards of the day, or else 'Marie' or 'Anne' which if you yelled it out in any deserted field would bring forth half a dozen women from toddler to granddam.
Moving on to the fretful topic of
In late C18, France spoke a dozen, mutally unintelligible dialects, which must have been interesting for all concerned.
'Annique' would be two syllables in Paris or in Brittany or Normandy -- pronounced exactly as you suggest. That's how Doyle would say it.
Somebody speaking the main sourthern dialect of French would make three syllables out of it, the southern types just generally adding the sound to the endings of words that north'rn folks chopped off.