We associate what we call vertex ℂ((z))-algebras and their modules in a certain category with elliptic affine Lie algebras. To a certain extent, this association is similar to that of vertex algebras and their modules with affine Lie algebras. While the notion of vertex ℂ((z))-algebra is a special case of that of quantum vertex ℂ((z))-algebra, which was introduced and studied by one of us (Li), here we use those results on quantum vertex ℂ(z))-algebras in an essential way. In the course of this work, we also construct and exploit two families of Lie algebras which are closely related to elliptic affine Lie algebras.