  Let $p \ge 5$ be a prime and let $\mathbb{Q}_{n,p}$ denote the $n$-th layer
of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. We show that
$\mathbb{Q}_{n,p}$ has no exceptional units. We use this to prove the effective
asymptotic Fermat's Last Theorem over $\mathbb{Q}_{n,p}$ for all $n \ge 1$ and
all primes $p \ge 5$ that are non-Wieferich, i.e. $2^{p-1} \not \equiv 1
\pmod{p^2}$. The effectivity in our result builds on recent work of Thorne
proving modularity of elliptic curves over $\mathbb{Q}_{n,p}$.
