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

On non-Archimedean Fréchet spaces with nuclear Köthe quotients