On non-Archimedean Fréchet spaces with nuclear Köthe quotients

Assume that $\mathbb{K}$ is a complete non-Archimedean valued field. We prove that every infinite-dimensional Fréchet-Montel space over $\mathbb{K}$ which is not isomorphic to $\mathbb{K}^{\mathbb{N}}$ has a nuclear Köthe quotient. If the field $\mathbb{K}$ is non-spherically complete, we show that every infinite-dimensional Fréchet space of countable type over $\mathbb{K}$ which is not isomorphic to the strong dual of a strict $LB$-space has a nuclear Köthe quotient.