You might be reading more than I intended into my previous response, which was really only a report of the help output and numerical precision. It is useful though to take a close at your data, and we will never recommend otherwise. If you need help with the analysis, we will need at least the exact command you used to start. It may be, in fact, that your results are not anomalous. I will say, however, that we tend to interpret the statistical test as a threshold, and not as the main result. This is in contrast to other software packages. We will generally look at beta coefficients, interpretable as percent signal change in our recommended pipelines for FMRI in afni_proc.py. So whether a p-value is less than some very small number or another infinitesimal number is less important than the effect size. Others may disagree.

[doi.org]

For some experiments, the model fits very well. Our class data includes an example for a single subject where the threshold does reach these kinds of heights. That data is for a somatosensory task, and the EPI data is an excellent fit. For your data, you may want to look closely at the underlying data and the fit to the model to see if it does indeed fit well. For any particular voxel, without taking into account clustering, you can take the data from all the individual subjects and analyze those with a separate statistical software to verify the basic t-test.