  We extend the well-known criterion of Lotz for the dual Radon-Nikodym
property (RNP) of Banach lattices to finitely generated Banach $C(K)$-modules
and Banach $C(K)$-modules of finite multiplicity. Namely, we prove that if $X$
is a Banach space from one of these classes then its Banach dual $X^\star$ has
the RNP iff $X$ does not contain a closed subspace isomorphic to $\ell^1$.
