<p>This paper presents strong second-order necessary and sufficient optimality conditions for the existence of a local Geoffrion proper efficient solution for a nonsmooth vector optimization problem. Primal form of second-order necessary optimality conditions is given for a problem with equality, inequality and abstract set constraints. The objective and inequality constraint functions are assumed to be locally Lipschitz and the equality constraint functions are assumed to be twice Fréchet differentiable where all the functions are defined on a Banach space. Dual form of second-order necessary optimality is derived for a problem without abstract set constraint in a finite dimensional setting assuming all the functions to be Fréchet differentiable. In the same finite dimensional setting second-order sufficient optimality is established assuming a calmness condition on the Fréchet derivatives of the involved functions.</p>