Weakly compact operators into sequence spaces: A counterexample

In a recent paper Gutiérrez and Villanueva have used, without giving a detailed proof, an analogue of a well-known result of Ryan characterizing the weakly compact operators from a Banach space $E$ into the space $c_0(X)$ of null sequences in a Banach space $X$. In this note a counterexample is given showing that in the statement of Gutiérrez and Villanueva an additional condition is needed.