Let $V$ and $W$ be two complex vector spaces with an action of a finite group $G$. The $G$-equivariant polynomial maps from $V$ to $W$ are finitely generated as a module over the ring of $G$-invariant polynomials on $V$. In other words, there exist $G$-equivariant polynomial maps $p_1,\dotsc,p_n$ so that any $G$-equivariant polynomial map may be written as $$q_1p_1+\dotsb +q_np_n$$ where the $q_i$ are $G$-invariant polynomials on $V$.

Is it true that the $G$-equivariant holomorphic maps $V\longrightarrow W$ are finitely generated as a module over the ring of $G$-invariant holomorphic functions on $V$, and that the generators may be taken as $G$-equivariant polynomial maps? In other words, may we also write any $G$-equivariant holomorphic map as $$f_1p_1+\dotsb +f_np_n$$ where the $f_i$ are now $G$-invariant holomorphic functions?