Suppose $R$ is a (commutative unital) topological ring which is admissible in the sense of Stacks 07E8: it is complete, Hausdorff, admits a fundamental system of neighborhoods of zero consisting of ideals, and admits an open ideal $I$ such that every neighborhood of 0 contains $I^n$ for some $n$. Is $R$ necessarily compactly generated as a topological space? What if we require $R$ to be Noetherian?
If $R$ is first-countable, e.g. if $R$ is adic (again in the sense of Stacks 07E8), the answer is yes: every first-countable space is compactly generated. However, I don't see any further cases.
I ask this in the hope that it is nicer to consider formal schemes to have a structure sheaf of condensed rings, rather than topological rings.