In this paper we study skew Poincaré–Birkhoff–Witt extensions over weak zip rings. Since these extensions generalize Ore extensions of injective type and another noncommutative rings of polynomial type, we unify and extend several results in the literature concerning the zip property. Under adequate conditions, we transfer this property from a ring of coefficients to a skew Poincaré–Birkhoff–Witt extension over this ring. We illustrate our results with examples of noncommutative algebras appearing in noncommutative algebraic geometry and theoretical physics.