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The aim of this work is to introduce a proper ε-subdifferential in the sense of Benson for extended mappings between topological linear spaces. It is defined by means of a new concept of a proper ε-efficient solution of a vector optimization problem. The main results are hold by assuming a sort of nearly subconvexlikeness, which is a very general concept of convexity. Some sufficient conditions for the existence of proper ε-subgradients are obtained. Our results generalize other similar results in the literature, and improve them since the limit behaviour when the error ε tends to zero of the involved concepts is more suitable.