Examples of non-archimedean nuclear Frécheat spaces without a Schauder basis

We solve the problem of the existence of a Schauder basis in non-archimedean Fréchet spaces of countable type (stated in [3]). Using examples of real nuclear Fréchet spaces without a Schauder basis (of Bessaga [1]), Mitiagin [5] and Vogt [10]) we construct examples of non-archimedean nuclear Fréchet spaces without a Schauder basis (even without the bounded approximation property).