  Let $K$ be a $p$-adic field, and let $K_\infty/K$ be a Galois extension that
is almost totally ramified, and whose Galois group is a $p$-adic Lie group of
dimension $1$. We prove that $K_\infty$ is not dense in
$(\mathbf{B}_{\mathrm{dR}}^+ / \operatorname{Fil}^2 \mathbf{B}_{\mathrm{dR}}^+
)^{\operatorname{Gal}(\overline{K}/K_\infty)}$. Moreover, the restriction of
$\theta$ to the closure of $K_\infty$ is injective, and its image via $\theta$
is the set of vectors of $\widehat{K}_\infty$ that are $C^1$ with zero
derivative for the action of $\operatorname{Gal}(K_\infty/K)$. The main
ingredient for proving these results is the construction of an explicit lattice
of $\mathcal{O}_{K_\infty}$ that is commensurable with
$\mathcal{O}_{K_\infty}^{d=0}$, where $d : \mathcal{O}_{K_\infty} \to
\Omega_{\mathcal{O}_{K_\infty} / \mathcal{O}_K}$ is the differential.
