Can a very small piece of material be superconducting? The existing theory of superconducting seems to be based on statistical mechanics.
Can an ultrasmall piece of material, like a quantum dot with very few atoms (like a small molecule), be superconducting?
For example, can a cubic of 3 * 3 * 3 = 27 copper atoms be superconducting?
What is the minimum n for a cubic of $n*n*n$ copper atoms to be able to be superconducting?
Can a few unit cells of a complex high temperature superconducting material be superconducting?
If so, then maybe some calculation from first principles can be done on such a piece of material as a molecule to understand the exact mechanisms of high temperature superconducting.
If not, can some first principle calculation on such a small piece of material be done to find some pattern that lead to a possible theory of high temperature superconducting?
 A: Tunneling of Cooper pairs through metallic islands has been a subject of research fir quite some time now. One does not call these quantum dots, since the QDs are usually understood in the context of semiconductors, but many ideas are the same - particularly relevant here is the Coulomb blockade, which permits counting electrons/pairs.
On the one hand, the operator of the particle number (i.e. the number of the Cooper pairs) does not commute with the semiconductor phase
$$[\varphi, N] = 1,$$ i.e. the Coulomb blockade destroys the superconductivity.
On the other hand, when the tunneling is possible, the number of particles on the island is not fixed, and the superconductivity does occur. But this is because the small volume in question is not really isolated.
To summarize


*

*Once the system is reduced to a countable number of particles (as suggested in the question), the superconductivity is impossible

*One can relieve this constraint by coupling the system to the environment, and thus inducing particle fluctuations.

*I would like to stress that this answer is complimentary to the one by @fra_pero, who focuses more on the long-range order and the boundary effects. 

A: First you need to define what "being superconducting" means for a finite-size (small) system.
From a theoretical point of view, superconductivity (as many other broken-symmetry phases) is defined in terms of long-range order, that is some correlation function $<\rho(r)\rho(0)>$ stays finite as $r$ goes to infinity. Then, for a finite-size system, one needs a criterion to decide how small it can be, in order for the concept of long-range order to make any sense.
From an experimental point of view, you probably need a system large enough to be able to perform a measurement of resistivity or of magnetic flux expulsion.
So the answer is yes, it can be superconducting, but it should not be too small. How small it can be, probably depends from case to case. A few unit cells, however, is for sure a too small system size for the concept of superconductivity (or any other phase of matter) to make any sense.
A: My experience in this field is mostly applied. Here is what i have seen in papers. Superconductors behave as 'macroscopic' only as long as their size is above the coherence length $\xi_0$. For example, in titanium this is nearly 0.5um, in niobium it is 20nm, in YBCO it is at the atomic scale (but anisotropic). 
Coherence length depends on temperature and applied magnetic field. 
When the size of a piece of superconducting material is decreased below its coherence length, one gets decrease in the critical temperature and critical magnetic field. 
To get tunneling of charge carriers in Josephson junctions their 'thickness' must be comparable or below the coherence length
