Is buckminsterfullerene aromatic?
Aromaticity is not binary, but rather there are degrees of aromaticity.  The degree of aromaticity in benzene is large, whereas the spiro-aromaticity in [4.4]nonatetraene is relatively small. The aromaticity in naphthalene is not twice that of benzene.

Aromaticity has come to mean a stabilization resulting from p-orbital (although other orbitals can also be involved) overlap in a pi-type system.  As the examples above indicate, the stabilization can be large or small.

Let's consider $\ce{C_{60}}$:


Bond alternation is often taken as a sign of non-aromatic systems.  In $\ce{C_{60}}$ there are different bond lengths, ~1.4 and 1.45 angstroms. However, this variation is on the same order as that found in polycyclic aromatic hydrocarbons, and less than that observed in linear polyenes.


Conclusion: aromatic, but less so than benzene.


Magnetic properties are related to electron delocalization and are often used to assess aromaticity.  Both experiment and calculations suggest the existence of ring currents (diamagnetic and paramagnetic) in $\ce{C_{60}}$.  


Conclusion: Although analysis is complex, analysis is consistent with at least some degree of aromaticity.


Reactivity - Substitution reactions are not possible as no hydrogens are present in $\ce{C_{60}}$.  When an anion or radical is added to $\ce{C_{60}}$ the electron(s) are not delocalized over the entire fullerene structure. However, most addition reactions are reversible suggesting that there is some extra stability or aromaticity associated with $\ce{C_{60}}$.


Conclusion: Not as aromatic as benzene


Resonance energy calculations have been performed and give conflicting results, although most suggest a small stabilization. Theoretical analysis of the following isodesmic reaction


$$\ce{C_{60} + 120 CH4 -&gt; 30 C2H4 + 60 C2H6}$$

suggested that it only took half as much energy to break all of the bonds in $\ce{C60}$ compared to the same bond-breaking reaction with the appropriate number of benzenes.

Conclusion: Some aromatic stabilization, but significantly less than benzene. 

This brief overview suggests that $\ce{C_{60}}$ does display properties that are consistent with some degree of aromatic stabilization, albeit less than that found with benzene.