Consider binary observations whose response probability is an unknown smooth function of a set of covariates. Suppose that a prior on the response probability function is induced by a Gaussian process mapped to the unit interval through a link function. In this paper we study consistency of the resulting posterior distribution. If the covariance kernel has derivatives up to a desired order and the bandwidth parameter of the kernel is allowed to take arbitrarily small values, we show that the posterior distribution is consistent in the $L_1$-distance. As an auxiliary result to our proofs, we show that, under certain conditions, a Gaussian process assigns positive probabilities to the uniform neighborhoods of a continuous function. This result may be of independent interest in the literature for small ball probabilities of Gaussian processes.