The ensemble Kalman filter is a computationally efficient technique to solve state and/or parameter estimation problems in the framework of statistical inversion when relying on a Bayesian paradigm. Unfortunately its cost may become moderately large for systems de- scribed by nonlinear time-dependent PDEs, because of the cost entailed by each PDE query. In this paper we propose a reduced basis ensemble Kalman filter technique to address the above problems. The reduction stage yields intrinsic approximation errors, whose propagation through the filtering process might affect the accuracy of state/parameter estimates. For an efficient evaluation of these errors, we equip our reduced basis ensemble Kalman filter with a reduction error model (or error surrogate). The latter is based on ordinary kriging for functional-valued data, to gauge the effect of state reduction on the whole filtering process. The accuracy and efficiency of the resulting method is then verified on the estimation of uncer- tain parameters for a FitzHugh-Nagumo model and uncertain fields for a Fisher-Kolmogorov model.