We study the class of Banach lattices that are positively polynomially Schur. Plenty of examples and counterexamples are provided, lattice properties of this class are proved, arbitrary
L
p
(
μ
)
L_p(\mu )
-spaces,
1
≤
p
>
∞
1 \leq p > \infty
, are shown to be positively polynomially Schur, lattice analogues of results on Banach spaces are obtained and relationships with the positive Schur and the weak Dunford-Pettis properties are established.