  Let $q$ be a prime power and $F=\mathbb{F}_q(T)$ be the rational function
field over $\mathbb{F}_q$, the field with $q$ elements. Let $\phi$ be a
Drinfeld module over $F$ and $\mathfrak{p}$ be a non-zero prime ideal of
$A:=\mathbb{F}_q[T]$. Over the constant $\mathbb{Z}_p$-extension of $F$, we
introduce the fine Selmer group associated to the $\mathfrak{p}$-primary
torsion of $\phi$. We show that it is a cofinitely generated module over
$A_{\mathfrak{p}}$. This proves an analogue of Iwasawa's $\mu=0$ conjecture in
this setting, and provides context for the further study of the objects that
have been introduced in this article.
