The main contribution is a result on the existence of an extension of a continuous multivalued selections that, under the assumption of completeness of the image space, generalizes the result of \textit{H. A. Antosiewicz} and \textit{A. Cellina} [Ann. Pol. Math. 34, 107-111 (1977; Zbl 0342.54008)]. Let E be a Banach space, \({\mathcal F}_ c(E)\) the family of all non-empty closed convex subsets of E, \({\mathcal F}_ b(E)\) the family of all non-empty closed bounded subsets of E. Theorem. Let X be a matric space, D a closed subset of X. Let F: \(X\to {\mathcal F}_ c(E)\) be lower semicontinuous, \(\Phi\) : \(D\to {\mathcal F}_ b(E)\) continuous and such that \(\Phi\) (x)\(\subset F(x)\) for all \(x\in D\). Then there exists a continuous extension \({\tilde \Phi}\): \(X\to F_ b(E)\) of \(\Phi\) such that \({\tilde \Phi}\)(x)\(\subset F(x)\) for any \(x\in X\).