We establish central limit theorems for a large class of supercritical branching Markov processes in infinite dimension with spatially dependent and non-necessarily local branching mechanisms. This result relies on a fourth moment assumption and the exponential convergence of the mean semigroup in a weighted total variation norm. This latter assumption is pretty weak and does not necessitate symmetric properties or specific spectral knowledge on this semigroup. In particular, we recover two of the three known regimes (namely the small and critical branching processes) of convergence in known cases, and extend them to a wider family of processes. To prove our central limit theorems, we use the Stein's method, which in addition allows us to newly provide a rate of convergence to this type of convergence.