Banach-Alaoglu Theorem over spherically complete non-Archimedean fields

Question

About a year ago I asked here whether the Banach-Alaoglu Theorem works over the $p$-adics. The satisfactory answer I got is that the "usual" proof only uses local compactness, and so the Banach-Alaoglu Theorem holds for any local field.

Now I would like to look at other, more general non-Archimedean fields. I know that Hahn-Banach holds for all spherically complete such fields, and so I was wondering if it is possible to prove Banach-Alaoglu for such fields as well? Because Hahn-Banach works, a related question is whether in the complex setting there is a proof of Banach-Alaoglu that uses Hahn-Banach, but not local compactness of $\mathbb{R}$ or $\mathbb{C}$.

Answer 1

In the general non-Archimedean case, instead of using the concept of compactness, it is more suitable to use the concept of compactoidness. I think the theorem you are looking for is

(page 273) all the details and preliminaries are in the spectacular book: Locally Convex Spaces over non-Arquimedean Valued Fields, Cambridge University Press - [C.Perez-Garcia, W.H.Schikhof] - 2010

Answer 2

Chilote has pointed to the right notion for the general case. I will answer the literal question.

The Banach-Alaoglu theorem (using the usual topological notion of compactness) cannot hold for normed spaces over a field with valuation $(k,|\cdot|)$ if the unit ball of $k$ is noncompact. The reason is that the weak-* dual of $k$ is isomorphic to $k$ with its original topology.

The spherical completion of the algebraic closure of $\mathbb{Q}_p$, for $p$ a prime, is a spherically complete field whose unit ball is noncompact.