Space has a smallest *measurable* unit, predicted at the planck length, but this is already far smaller than our ability to observe. A quantum theory of gravity may yield a more fundamental unit in space, but it's worth noting that we derive the Planck length from dimensional analysis, and as such, it is mute on numerical factors, which we need further exacting theories to describe, and we already know of effects with less length than a planck length. The gravitational effect of an electron, which must exist, as it has mass, extends orders of magnitude less than the Planck length. Of course, we have no quantum theory of gravity, yet, and many hypothesis use the Planck scale as a foundational part of the hypothesis, but I don't think you're bold enough to tell me that electrons do not possess any gravitational force.

What Hitsumei said is right, though, what I said does not logically follow if you are unwilling to describe space as pure geometry. But space looks a lot like pure geometry, uncertainties of position and distance being the very reason we use a fundamental unit. It says as much in the "Physical Signifigance" section of the Wiki article on Planck length -- that our ability to measure anything at less than the Planck length is very much in doubt, not doubting that such positions and distances exist. Indeed, if they did not, you and I would not be able to move, as the Planck length is the distance covered by a photon in the Planck time, and an electron or proton do not move at the speed of light. Indeed, all movement would be forced to shift in frames. Time doesn't appear to operate in slices, though, it just gets very hard to tell when it is progressing at all.

Note that nowhere did I claim we could stuff an infinite amount of particles, nor an infinite amount of equal finite spaces, into any given space, only that an infinite number of points may be defined. Space is foamy, and while you cannot define below a Planck length in distance and position relative to something else and then measure it, you can most certainly center any particle on any point in space. As it is, experimental inaccuracies prevent us from verifying if that particle is centered on a point less than a Planck distance from some other object (actually experimental inaccuracies prevent us from doing measurements at several magnitudes higher, at the moment, but you understand). I say again, such points ought to exist if reality actually consists of geometric spatial dimensions.