I want to know an example of $\Bbb{Z}_p$-extension over $\Bbb{Q}_p$.
$\Bbb{Z}_p$-extension over $K$ is defined as galois extension over $K$ whose galois group is isomorphic to $\Bbb{Z}_p$ as an abelian group.
In the case $K＝\Bbb{Q}$, $\Bbb{Z}_p$-extension is uniquely determined and which is called cyclotomic $\Bbb{Z}_p$-extension.
But what about in the case of $K＝\Bbb{Q}_p$? Is there $\Bbb{Z}_p$-extension over $\Bbb{Q}_p$ ?
One can easily generalize the construction of the cyclotomic $\mathbb Z_p$-extension over $\mathbb Q_p$. This is the case the local field $\mathbb Q_p$ does not interfer with $p$-th power roots of unity (the only new roots of unity are the $(p-1)$-th roots of unity by Hensel's lemma).
Indeed, the extensions $K_n=\mathbb Q_p(\zeta_{p^n})$ are still Galois extension with groups $(\mathbb Z/p^n\mathbb Z)^\ast$. A case distinction for $p=2$ and $p\ne2$ yields a sequence of compatible extension fields with Galois groups $\mathbb Z/p^n\mathbb Z$, hence a $\mathbb Z_p$-extension of $\mathbb Q_p$ by the general theory of infinite Galois groups.