Pop-sci ST books say, "At every point of space, there is a Calabi-Yau space..." or similar. But uncertainty p. and noncommgeom say, no points in space, only planckons (not my coinage). Is 'point of space' (meaning real, not abstract/mathematical space) inexact terminology, or a category error, like "an atom of mayonnaise'- only worse? So I speculate that C-Y space can not be a continuum either, but must consist of discrete calabicles (my coinage).
The Calabi-Yau space description is an approximation, valid when the manifold is big compared to the Planck scale, and the string scale. The string scale is about 10-100 times bigger than the Planck scale, and the CY is about 500-5000 times bigger.
At the string scale, the manifold stays a manifold, geometry still makes sense, because traditional superstrings strings move in a classical background geometry. But at the Planck length, there is no sense in saying that it stays a manifold. But it doesn't break up into points.
The measurable things about the CY are encoded in the effect it has on the physics of the large dimension, on the scattering information that can get to infinity to be measured by us. When the manifold is this small, its geometry is not 100% well defined. For example, in type II string theories, you can find Mirror-symmetry, that two completely different Calabi Yau's are physically exactly the same, but for the two different type II string theories. So which one of the two Calabi-Yaus is really attached at this point? The question has no meaning.
Any atoms of space-time are not to be thought of as living in the space time, but on holographic boundaries. If you want the closest thing we have to "Calabicles", look at the description of string theory in terms of D0 branes called Matrix theory. The pop-sci versions of the story tend to underemphasize the degree to which string theory is logically positivistic, only allowing measurements of space-time at asymptotic surfaces, and giving no meaning to its microstructure. String theory is the most positivistic theory ever constructed.