What is the smallest object that can spin?

Yes - protons have a so-called "spin quantum number". It is 1/2 for protons. Protons are made of quarks, which also have a 1/2 spin quantum number. In fact all known fermions (which are what make up ordinary matter) have spin 1/2. Spin 1/2 means the particle must be fully rotated twice to end up in its original configuration.

You can read more about spin quantum numbers on the Wikipedia article for spin-1/2.

If what you meant by smallest particles was particles having the smallest mass, one can list the followings. But before doing so, bear in mind that unlike what you asked in the question, spin doesn't exactly correspond to anything familiar from the classical mechanics.

Think about an electron and it has spin $pm1/2$, since it's an elementary particle and have no internal structure they are point-like and it's not meaningful to define classical rotation of a point-like particle around its own axis (try to imagine a point rotating around it's own axis, you'll get what I mean). Spin, unlike classical rotation, is an intrinsic form of angular momentum.

Some of the smallest particles and their spins (referring directly to your question): , check out photon, gluon, electron neutrino and up quark.