Is it possible to have a time-dependent (non-conservative) force satisfying $\nabla \times \vec{F}=0$?

Yes, in general. The two conditions for a force to be conservative-- that it depends only on $\vec{r}$, and that $\nabla \times \vec{F}=0$ (the latter is equivalent to the condition that work done is independent of path) -- are separate conditions, and if either one is violated, then the force is non-conservative.