Every non-normable non-archimedean Köthe space has a quotient without the bounded approximation property

It is proved that any non-archimedean non-normable Fréchet space with a Schauder basis and a continuous norm has a quotient without the bounded approximation property. It follows that any infinite-dimensional non-archimedean Fréchet space, which is not isomorphic to any of the following spaces: , has a quotient without a Schauder basis. Clearly, any quotient of c₀ and has a Schauder basis. It is shown a similar result for and