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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 thereexists a number field $L$ and a $\mathbf{Z}_p$ extension $L_{\infty}/L$ wherethe $\ell$-primary fine Selmer group of $A$ grows arbitrarily quickly. This isa fine Selmer group analogue of a theorem of Washington which says that thereare certain (non-cyclotomic) $\mathbf{Z}_p$-extensions where the $\ell$-part ofthe class group can grow arbitrarily quickly. We also prove this for a wideclass of non-commutative $p$-adic Lie extensions. Finally, we include severalexamples to illustrate this theorem.

The non-$p$-part of the fine Selmer group in a $\mathbf{Z}_p$-extension