We introduce and study Banach lattices with the strong Dunford-Pettis relatively compact property of order p (1 ≤ p < ∞); that is, spaces in which every weakly p-compact and almost Dunford-Pettis set is relatively compact. We also introduce the notion of the weak Dunford-Pettis property of order p and then characterize this property in terms of sequences. In particular, in terms of disjoint weakly compact operators into c 0 , an operator characterization of those Banach lattices with the weak Dunford-Pettis property of order p is given. Moreover, some results about Banach lattices with the positive Dunford-Pettis relatively compact property of order p are presented.