Are there any restrictions on the ground field over which the implicit function theorem holds? In particular, does the theorem hold over function fields like $F_q((1/t))$?

There is an implicit function theorem valid for any nonArchimedean field. See Theorem 2.2.1 in J.I. Igusa, An introduction to the theory of local zeta functions. AMS, 2000. 


Essentially, it is explained in the answer to this math.stackexchange question that the implicit function theorem is equivalent to the validity of Hensel's lemma. The fields where the implicit function theorem holds are called henselian fields. 

