  We show that the Frobenius--stable version of the Grauert--Riemenschneider
vanishing theorem fails for terminal 3-folds in characteristic 2. To prove
this, we introduce the notion of $\mathbb{F}_p$-rationality for singularities
in positive characteristic and we show that klt singularities in dimension at
most 4 are $\mathbb{F}_p$-rational.
