We consider a sensor network in which each sensor takes measurements, at various times, of some unknown parameters, corrupted by independent Gaussian noises. Each node can take a finite or infinite number of measurements, at arbitrary times (i.e., asynchronously). We propose a space-time diffusion scheme, that relies only on peer-to-peer communication, and allows every node to asymptotically compute the global maximum-likelihood estimate of the unknown parameters. At each iteration, information is diffused across the network by a temporal update step and a spatial update step. Both steps update each node's state by a weighted average of its current value and locally available data: new measurements for the time update, and neighbors' data for the spatial update. At any time, any node can compute a local weighted least-squares estimate of the unknown parameters, which converges to the global maximum-likelihood solution. With an infinite number of measurements, these estimates converge to the true parameter values in the sense of mean-square convergence. We show that this scheme is robust to unreliable communication links, and works in a network with dynamically changing topology.