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Abstract. Given a subset A of a topological space X, a locally convex space Y, and a family C of subsets of Y we study the problems of the existence of a linear C-extender u: C∞(A, Y) → C∞(X,Y), which is a linear operator extending bounded continuous functions f: A → C ⊂ Y, C ∈ C, to bounded continuous functions f ̄ = u(f) : X → C ⊂ Y. Two necessary conditions for the existence of such an extender are found in terms of a topological game, which is a modification of the classical strong Choquet game. The obtained results allow us to characterize reflexive Banach spaces as the only normed spaces Y such that for every closed subset A of a GO-space X there is a C-extender u: C∞(A, Y) → C∞(X,Y) for the family C of closed convex subsets of Y. Also we obtain a characterization of Polish spaces and a characterization of weakly sequentially complete Banach lattices in terms of extenders