Full-history recursive multilevel Picard (MLP) approximation schemes have been shown to overcome the curse of dimensionality in the numerical approximation of high-dimensional semilinear partial differential equations (PDEs) with general time horizons and Lipschitz continuous nonlinearities. However, each of the error analyses for MLP approximation schemes in the existing literature studies the $L^2$-root-mean-square distance between the exact solution of the PDE under consideration and the considered MLP approximation and none of the error analyses in the existing literature provides an upper bound for the more general $L^p$-distance between the exact solution of the PDE under consideration and the considered MLP approximation. It is the key contribution of this article to extend the $L^2$-error analysis for MLP approximation schemes in the literature to a more general $L^p$-error analysis with $p\in (0,\infty)$. In particular, the main result of this article proves that the proposed MLP approximation scheme indeed overcomes the curse of dimensionality in the numerical approximation of high-dimensional semilinear PDEs with the approximation error measured in the $L^p$-sense with $p \in (0,\infty)$.