The purpose of the present paper is to prove the mutual equivalence of some boundedness properties of subsets of non-archimedean (n. a.) Banach spaces. In particular, the boundedness by norm is equivalent to the weak boundedness by functional. We consider some properties of bounded subsets of n. a. vector spaces and their operator rings. For example, we give an algebraic characterization of bounded subsets in the operator rings of n. a. vector spaces and, as a corollary, we obtain an axiomatic description of the operator norm.