This work concerns state-space models, in which the state-space is an infinite-dimensional spatial field, and the evolution is in continuous time, hence requiring approximation in space and time. The multilevel Monte Carlo (MLMC) sampling strategy is leveraged in the Monte Carlo step of the ensemble Kalman filter (EnKF), thereby yielding a multilevel ensemble Kalman filter (MLEnKF) for spatio-temporal models, which has provably superior as- ymptotic error/cost ratio. A practically relevant stochastic partial differential equation (SPDE) example is presented, and numerical experiments with this example support our theoretical findings.