Examples of non-archimedean Fréchet spaces without nuclear Köthe quotients

Let K be a spherically complete non-archimedean valued field. We prove that the dual space l∞ of the Banach space c0 has a total strongly non-norming subspace M. Using this subspace M we construct a non-normable Fréchet space F of countable type with a continuous norm such that its strong dual is a strict LB-space. Next we show that F has no nuclear Köthe quotient.