Let K be a non-trivial complete non-Archimedean valued field and let E be an infinite-dimensional Banach space over K. Some of the main results are:
(1) K is spherically complete if and only if every weakly convergent sequence in l∞ is norm-convergent.
(2) If the valuation of K is dense, then C0
is complemented in E if and only if C(E,c0) is n o t complemented in L(E,c0), where L(E,c0) is the space of all continuous linear operators from E to c0
and C(E,c0) is the subspace of L(E, c0) consisting of all compact linear operators.