It is a challenging topic in applied mathematics to solve high-dimensional nonlinear partial differential equations (PDEs). Standard approximation methods for nonlinear PDEs suffer under the curse of dimensionality (COD) in the sense that the number of computational operations of the approximation method grows at least exponentially in the PDE dimension and with such methods it is essentially impossible to approximately solve high-dimensional PDEs even when the fastest currently available computers are used. However, in the last years great progress has been made in this area of research through suitable deep learning (DL) based methods for PDEs in which deep neural networks (DNNs) are used to approximate solutions of PDEs. Despite the remarkable success of such DL methods in simulations, it remains a fundamental open problem of research to prove (or disprove) that such methods can overcome the COD in the approximation of PDEs. However, there are nowadays several partial error analysis results for DL methods for high-dimensional nonlinear PDEs in the literature which prove that DNNs can overcome the COD in the sense that the number of parameters of the approximating DNN grows at most polynomially in both the reciprocal of the prescribed approximation accuracy $\varepsilon>0$ and the PDE dimension $d\in\mathbb{N}$. In the main result of this article we prove that for all $T,p\in(0,\infty)$ it holds that solutions $u_d\colon[0,T]\times\mathbb{R}^d\to\mathbb{R}$, $d\in\mathbb{N}$, of semilinear heat equations with Lipschitz continuous nonlinearities can be approximated in the $L^p$-sense on space-time regions without the COD by DNNs with the rectified linear unit (ReLU), the leaky ReLU, or the softplus activation function. In previous articles similar results have been established not for space-time regions but for the solutions $u_d(T,\cdot)$, $d\in\mathbb{N}$, at the terminal time $T$.