"She is Anne."
No. The basic rule is that you must use ce when être is followed by any determiner—for instance, an article or a possessive adjective.
Note that c'est should be used for singulars and ce sont should be used for plurals.
This rule also applies to proper nouns (as in people's name's) that directly follow être, such as Anne.
C'est un homme. — He's a man. / This is a man. / That is a man.
Ce sont des chats. — They're cats. / These are cats. / Those are cats.
C'est mon chien. — It's my dog. / This is my dog. / That's my dog.
And C'est Anne → She / This / That is Anne.
"il/elle est" and "ils/elles sont" change to "c'est" and "ce sont" before a modified noun, that is, a noun preceded by a modifier. A modifier can be:
- an article: un, une, des, le, la, l', les
- a number: un, deux...
- a possessive adjective: mon, ton, son, ma, ta, sa, notre, votre, leur, mes, tes, ses, nos, vos, leurs
- a demonstrative adjective: ce, cet, cette, ces
The rule also applies to names - you can't say "Elle est Anne" or "Il est Bruno", you use "C'est Anne" and "C'est Bruno".
These articles go into more detail and are worth a read.
Also, the Tips and Notes for the Gallicism skill (click the lightbulb icon when you open the skill) give more information.