Nic

Also a Romansh short form of Nicolà.
Source: "Vornamen in der Schweiz. Prénoms en Suisse. I nomi in Svizzera. Prenoms in Svizra" (1993) published by the Association of Swiss registrars
https://sursassiala.ch/2015/01/15/familienforschung/

Also a Jèrriais short form of Nicolas: https://www.learnjerriais.org.je/pluginfile.php/3125/mod_resource/content/1/dictionnaithe%20Angliais-J%C3%A8rriais%202012.pdf
http://members.societe-jersiaise.org/geraint/jerriais/noms1ers.html

Can't this also be a diminutive of Nicolette?

In modern times, Nic is also used as a feminine name, as a diminutive of Nicole or Dominique.

A friend of mine named Nicole uses this nickname.