Locally analytic vectors and $\mathbf{Z}_p$-extensions

Let $K$ be a finite extension of $\mathbf{Q}_p$ and let $\mathcal{G}_K = \mathrm{Gal}(\overline{\mathbf{Q}_p}/K)$. Lately, interest has risen around a generalization of the theory of $(\varphi,\Gamma)$-modules, replacing the cyclotomic extension with an arbitrary infinitely ramified $p$-adic Lie extension. Computations from Berger suggest that locally analytic vectors should provide such a generalization for any arbitrary infinitely ramified $p$-adic Lie extension, and this has been conjectured by Kedlaya. In this paper, we focus on the case of $\mathbf{Z}_p$-extensions, using recent work of Berger-Rozensztajn and Porat on an integral version of locally analytic vectors and explain what can be the structure of the locally analytic vectors in the higher rings of periods $\widetilde{\mathbf{A}}^{\dagger}$ in this setting. We show that the existence of nontrivial locally analytic vectors in $\widetilde{\mathbf{A}}^{\dagger}$, a necessary condition for Kedlaya's conjecture to hold, is equivalent to the existence of an overconvergent lift of the field of norms attached to the $\mathbf{Z}_p$-extension. In the anticyclotomic setting, assuming that such an overconvergent lift exists, we are able to construct elements in the corresponding Robba ring which should not exist according to a conjecture of Berger. We then prove that in this specific setting, a particular case of Berger's conjecture holds, discarding the existence of such elements. In particular, this disproves Kedlaya's conjecture and shows that there is no overconvergent lift of the field of norms in the anticyclotomic setting.