A stochastic gravitational-wave background (SGWB) can arise from the superposition of many independent events. If the fraction of time that events emit in some frequency band is sufficiently high, the resulting background is Gaussian in that band, which is to say that it is characterized only by a gravitational-wave strain power spectrum. Alternatively, if this fraction is low, we expect a non-Gaussian background, characterized by intermittent subthreshold signals. Many experimentally accessible models of the SGWB, such as the SGWB arising from compact binary coalescences, are expected to be of this non-Gaussian variety. Primordial backgrounds from the early Universe, on the other hand, are more likely to be Gaussian. Measuring the Gaussianity of the SGWB can therefore provide additional information about its origin. In this paper we introduce a novel maximum likelihood estimator that can be used to estimate the non-Gaussian component of an SGWB signature measured in a network of interferometers. This method can be robustly applied to spatially separated interferometers with colored, non-Gaussian noise. Furthermore, it can be cast as a generalization of the widely used stochastic radiometer algorithm.