The primary aim of this paper is to develop the theory of product local Hardy spaces on Banach lattices. We begin by introducing local Hardy spaces associated with product ball quasi-Banach function spaces, denoted as \(h_X(\mathbb {R}^n \times \mathbb {R}^m)\) , using the Littlewood-Paley-Stein theory. Subsequently, we establish the boundedness of bi-parameter inhomogeneous singular integral operators on \(h_X(\mathbb {R}^n \times \mathbb {R}^m)\) by applying the discrete local Calderón reproducing formula and the Littlewood–Paley–Stein theory. This requires only the mild assumptions that the vector-valued maximal inequality holds for \(X(\mathbb {R}^n \times \mathbb {R}^m)\) , and that \(X(\mathbb {R}^n \times \mathbb {R}^m)\) possesses an absolutely continuous quasi-norm. To include spaces that do not have an absolutely continuous quasi-norm, we utilize the extrapolation theory to prove the \((h_X(\mathbb {R}^n \times \mathbb {R}^m), X(\mathbb {R}^n \times \mathbb {R}^m))\) boundedness and the \((h_X(\mathbb {R}^n \times \mathbb {R}^m), h_X(\mathbb {R}^n \times \mathbb {R}^m))\) boundedness of bi-parameter inhomogeneous singular integral operators and bi-parameter pseudo-differential operators, which needs an additional condition that the strong Hardy-Littlewood maximal operator \(\mathcal {M}_{s}\) is bounded on the associate space of the convexification of \(X(\mathbb {R}^n \times \mathbb {R}^m)\) . Finally, we apply these results to two concrete examples of ball quasi-Banach function spaces, including product Herz spaces and weighted product Morrey spaces.
