We deal with classes of normed spaces E over a non-archimedean valued field for which every one-dimensional (and hence every finite-dimensional) subspace satisfies some orthocomplementation property. They are described in terms of polarity for the balls of E and in terms of compactoidity for the balls of E'. This study yields the solution of an open problem raised by A.C.M. van Rooij and W.H. Schikhof in [Open problems. In: p-Adic Functional Analysis, 209-219, Lecture Notes in Pure and Appl. Math., 137, Marcel Dekker, New York, 1992].