It is proved that the following conditions are equivalent: (i) \(X\) is normal and countably paracompact; (ii) For every convex subset \(C\) of a separable Banach space \(B\), every lsc multifunction \(F: X\to C\) with values convex and compact in \(B\) and every \(F_\sigma\)-subset of \(X\) contains in \(F^-(\text{Int\,}C)\) there exists a continuous selection \(f\) for \(F\) with \(A\subset f^{-1}(\text{Int\,}C)\subset F^-(\text{Int\,}C)\); (iii) For every \(F: X\to C\) with closed and convex values in \(B\) the thesis of (ii) holds. If \(X\) is paracompact space, then a similar selection theorem without separability assumptions imposed on \(B\) is true. As a corollary some results concerning extensions of products and of disjoint families of single-valued functions are obtained. A counterexample giving a solution to a question raised in [\textit{M. Frantz}, Pac. J. Math. 169, No. 1, 53--73 (1995; Zbl 0843.54024)] is also constructed.