We propose a Kalman filter algorithm to provide a formal statistical analysis of space-time data with an autoregressive structure in time. The Kalman filter technique allows to capture the temporal dependence as well as the spatial correlation structure through state-space equations, and it is aimed to perform statistical inference in terms of parameter estimation and prediction at unobserved locations. We thus develop space-time estimation and prediction methods in the presence of missing data, through the Kalman filter, in order to obtain accurate estimates of model parameters and reliable space-time predictions. Our findings are illustrated through an application on daily air temperatures in some regions of southern Chile, where the dataset shows a number of missing data in many locations.