  Fix two distinct primes $p$ and $\ell$. Let $A$ be an abelian variety over
$\mathbf{Q}(\zeta_{\ell})$, the cyclotomic field of $\ell$-th roots of unity.
Suppose that $A(\mathbf{Q}(\zeta_{\ell}))[\ell] \neq 0$. We show that there
exists a number field $L$ and a $\mathbf{Z}_p$ extension $L_{\infty}/L$ where
the $\ell$-primary fine Selmer group of $A$ grows arbitrarily quickly. This is
a fine Selmer group analogue of a theorem of Washington which says that there
are certain (non-cyclotomic) $\mathbf{Z}_p$-extensions where the $\ell$-part of
the class group can grow arbitrarily quickly. We also prove this for a wide
class of non-commutative $p$-adic Lie extensions. Finally, we include several
examples to illustrate this theorem.
