We present approximate algorithms for performing smoothing in a class of high-dimensional state-space models via sequential Monte Carlo methods ("particle filters"). In high dimensions, a prohibitively large number of Monte Carlo samples ("particles"), growing exponentially in the dimension of the state space, is usually required to obtain a useful smoother. Employing blocking approximations, we exploit the spatial ergodicity properties of the model to circumvent this curse of dimensionality. We thus obtain approximate smoothers that can be computed recursively in time and parallel in space. First, we show that the bias of our blocked smoother is bounded uniformly in the time horizon and in the model dimension. We then approximate the blocked smoother with particles and derive the asymptotic variance of idealised versions of our blocked particle smoother to show that variance is no longer adversely effected by the dimension of the model. Finally, we employ our method to successfully perform maximum-likelihood estimation via stochastic gradient-ascent and stochastic expectation-maximisation algorithms in a 100-dimensional state-space model.