We prove that the Lorentz sequence spaces do not have the property B of Lindenstrauss. In fact, for any admissible sequences w, v ∈ c0 \ l1, the set of norm-attaining operators from the Orlicz space hϕ(w) (ϕ is a certain Orlicz function) into d(v, 1) is not dense in the corresponding space of operators. We also characterize the spaces such that the subset of norm-attaining operators from the Marcinkiewicz sequence space into its dual is dense in the space of all bounded and linear operators between them.