  Let $p$ be an odd prime and $F_\infty$ be a $\mathbb{Z}_p$-extension of a
number field $F$. Given an elliptic curve $E$ over $F$, we study the structure
of the fine Selmer group over $F_\infty$. It is shown that under certain
conditions, the fine Selmer group is a cofinitely generated module over
$\mathbb{Z}_p$ and furthermore, we obtain an upper bound for its corank (i.e.,
the $\lambda$-invariant), in terms of various local and global invariants.
