The logistic regression model has become a standard tool to investigate the relationship between a binary outcome and a set of potential predictors. When analyzing binary data, it often arises, however, that the observed proportion of zeros is greater than expected under the postulated logistic model. Zero-inflated binomial (ZIB) models have been developed to fit binary data that contain too many zeros. Maximum likelihood estimators in these models have been proposed, and their asymptotic properties were recently established. In this paper, we use these asymptotic properties to construct simultaneous confidence bands for the probability of a positive outcome in a ZIB regression model. Simultaneous confidence bands are especially attractive since they allow inference to be made over the whole regressor space. We construct two types of confidence bands, based on: (i) the Scheffé method for the linear regression model; (ii) Monte Carlo simulations to approximate the distribution of the supremum of a Gaussian field indexed by the regressor. The finite-samples properties of these two types of bands are investigated and compared in a simulation study.