In this work we consider estimation of the average causal effect in a model where the covariates required for confounding adjustment are discrete but arbitrarily high-dimensional. Surprisingly, no results are available yet in this setting. We first provide non-asymptotic risk bounds for a standard plug-in estimator, showing that this estimator is only consistent in the regime where the dimension grows slower than the sample size. Then we go on to characterize the minimax lower bound. We also consider several variations on this setup: one where we weaken the classical positivity assumption so that the bound on propensity scores can become more extreme with sample size (in which the functional becomes nonsmooth and the minimax rates change substantially), one where we instead target a data-adaptive causal effect conditional on being in a high-probability category, and one where we consider a sparsity condition that limits the heterogeneity of conditional causal effects.