  We consider a ballistic random walk in an i.i.d. random environment that does
not allow retreating in a certain fixed direction. We prove an invariance
principle (functional central limit theorem) under almost every fixed
environment. The assumptions are nonnestling, at least two spatial dimensions,
and a $2+\epsilon$ moment for the step of the walk uniformly in the
environment. The main point behind the invariance principle is that the
quenched mean of the walk behaves subdiffusively.
