On compactoidity in non-archimedean locally convex spaces with a Schauder basis

Several properties of compactoid sets in non-archimedean locally convex spaces with a Schauder basis are proved in this paper. As a consequence we derive partial affirmative answers to the questions formulated by Gruson and Van der Put ([4], problem following 5.8) and Schikhof ([6], problem following 1.7), respectively.