  Let $K$ be a number field, and $S$ a finite set of non-archimedean places of
$K$, and write $\mathcal{O}_S^\times$ for the group of $S$-units of $K$. A
famous theorem of Siegel asserts that the $S$-unit equation
$\varepsilon+\delta=1$, with $\varepsilon$, $\delta \in \mathcal{O}_S^\times$,
has only finitely many solutions. A famous theorem of Shafarevich asserts that
there are only finitely many isomorphism classes of elliptic curves over $K$
with good reduction outside $S$. Now instead of a number field, let
$K=\mathbb{Q}_{\infty,\ell}$ which denotes the $\mathbb{Z}_\ell$-cyclotomic
extension of $\mathbb{Q}$. We show that the $S$-unit equation
$\varepsilon+\delta=1$, with $\varepsilon$, $\delta \in \mathcal{O}_S^\times$,
has infinitely many solutions for $\ell \in \{2,3,5,7\}$, where $S$ consists
only of the totally ramified prime above $\ell$. Moreover, for every prime
$\ell$, we construct infinitely many elliptic or hyperelliptic curves defined
over $K$ with good reduction away from $2$ and $\ell$. For certain primes
$\ell$ we show that the Jacobians of these curves in fact belong to infinitely
many distinct isogeny classes.
