The iterative ensemble Kalman filter (IEnKF) was recently proposed in order to improve the performance of ensemble Kalman filtering with strongly nonlinear geophysical models. The IEnKF can be used as a lag-one smoother and extended to a fixed-lag smoother: the iterative ensemble Kalman smoother (IEnKS). The IEnKS is an ensemble variational method. It does not require the use of the tangent linear of the evolution and observation models, nor the adjoint of these models: the required sensitivities (gradient and Hessian) are obtained from the ensemble. Looking for optimal performance, out of the many possible extensions we consider a quasi-static algorithm. The IEnKS is explored for the Lorenz '95 model and for a two-dimensional turbulence model. As the logical extension of the IEnKF, the IEnKS significantly outperforms standard Kalman filters and smoothers in strongly nonlinear regimes. In mildly nonlinear regimes (typically synoptic-scale meteorology), its filtering performance is marginally but clearly better than the standard ensemble Kalman filter and it keeps improving as the length of the temporal data assimilation window is increased. For long windows, its smoothing performance outranks the standard smoothers very significantly, a result that is believed to stem from the variational but flow-dependent nature of the algorithm. For very long windows, the use of a multiple data assimilation variant of the scheme, where observations are assimilated several times, is advocated. This paves the way for finer reanalysis, freed from the static prior assumption of 4D-Var but also partially freed from the Gaussian assumptions that usually impede standard ensemble Kalman filtering and smoothing.