Is Ohm's Law always exactly true even at the microscopic level?
Consider the rule that the volume of liquid in a vessel is equal to the height of the liquid times its horizontal cross sectional area.  For vessels whose horizontal cross section is uniform from top to bottom, halving the height of the liquid will halve its volume.  Doubling the amount of liquid, if there's room, will double the volume.

Suppose, however, one removes nearly all of the liquid from the vessel, leaving only two nanoliters.  and then cuts the height of the liquid in half.  Should one expect that exactly one nanoliter would remain?

The relationship between cross-sectional area, height, and volume remains true by definition at any scale, large or small, provided that those quantities are concretely defined numbers.  At very small scales, however, concepts like "volume" become rather nebulous.  If there are only a few hundred molecules bouncing around, it's unclear how much of the space outside each should count as part of the "volume" of liquid, and how much should be considered "empty space".  In such cases, the relationship between height, volume, and cross sectional area doesn't "break down" so much as it becomes less and less meaningful, especially in cases where the cross-sectional area varies widely with height.

To put it another way, when quantities get very small, measurement uncertainties get very big.  When quantities get so small that measurement uncertainties get to be as big as the measurements, Ohm's Law will still hold, but will cease to have much predictive value.