We introduce a chain condition (bishop), defined for operators acting on
C(K)-spaces, which is intermediate between weak compactness and having weakly
compactly generated range. It is motivated by Pe{\l}czy\'nski's
characterisation of weakly compact operators on C(K)-spaces. We prove that if K
is extremally disconnected and X is a Banach space then an operator T : C(K) ->
X is weakly compact if and only if it satisfies (bishop) if and only if the
representing vector measure of T satisfies an analogous chain condition. As a
tool for proving the above-mentioned result, we derive a topological
counterpart of Rosenthal's lemma. We exhibit several compact Hausdorff spaces K for which the identity operator on C(K) satisfies (bishop), for example both
locally connected compact spaces having countable cellularity and ladder system
spaces have this property. Using a Ramsey-type theorem, due to Dushnik and
Miller, we prove that the collection of operators on a C(K)-space satisfying
(bishop) forms a closed left ideal of B(C(K)).