The Hahn-Banach subspaces of Banach spaces with base

Let K be a non-archimedean, complete valued field which is not spherically complete. We get a characterization of HB-subspaces in Banach spaces which possess an orthonormal base (all these Banach spaces are isometrically isomorphic to co(I) for some set I). A closed subspace G subset of c(0)(I) of countable type or such that c(0)(I)/G is of countable type is a Hahn-Banach subspace of c(0)(I) if and only if G is orthocomplemented in c(0)(I). This fact extends a similar result obtained earlier for c(0). Additionally, we observe that this theorem cannot be generalized for an arbitrary Banach space with (not necessary orthogonal) base.