The investigation of the dimension of Bergman spaces has long been a central topic in several complex variables, uncovering profound connections with potential theory and function theory since the pioneering work of Carleson, Wiegerinck, and others in the 1960s. We investigate the dimension of $p$-Bergman spaces associated with pseudoconvex domains in $\mathbb{C}^n$. By constructing $L^p$-versions of the extension theorems of Ohsawa and Ohsawa-Takegoshi, we establish several geometric and potential-theoretic criteria that ensure the spaces are infinite-dimensional. Sufficient conditions for the infinite dimensionality of $p$-Bergman spaces of complete N-circled fibered Hartogs domains, balanced domains, and weighted $p$-Fock spaces are obtained by applying the mentioned $L^p$-analogs of extension theorems and generalizing a sufficient condition of Jucha.