This question should be fairly elementary. It is related to this question, and I’d just like to check I’m not missing anything.

Let $\{M_n\}_{n\ge 0}$ be an inverse system of smooth manifolds, resp smooth complex analytic spaces, with transition maps $f_{t,s} : M_t\to M_s$, $t\ge s$, that are local diffeomorphisms, resp local analytic isomorphisms.

Let $M$ be the topological inverse limit.

For every $x\in M$, does there exist an open neighborhood $U\subset M$ of $x$, such that $U$ is homeomorphic to some smooth manifold, resp smooth complex analytic space, that is diffeomorphic, resp locally analytically isomorphic, to an open subset of $M_n$ for some $n$?

This is certainly true for each of the finite layers, and should be true in the limit. In particular, $M$ would be locally compact Hausdorff and second countable, and one may define a notion of “smooth” partition of unity, and give a sufficient answer to the linked question.