For example, $0.1c < v < c$? Is this actually possible?
Background: is that true that the black hole in the center of the Milky Way rotates once per 20 minutes?
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A particle travelling in a circle of radius $R$ at velocity $v$ has an acceleration of $v^2/R$. So either $R$ is very large, or a very strong force is acting to stop the particles at the surface of the object flying away. We see no very large objects (galactic clusters, for instance) rotating this quickly, so we need to look for strong forces.
One option is gravity. We know of sub-millisecond pulsars, which we believe are neutron stars with a radius of perhaps 10 km, rotating 1000+ times per second. This gives a surface velocity at the equator of about $60\,000 km/s$ which is $0.2c$. So those meet your conditions.
Defining how fast the surface of a black hole is rotating is difficult on several levels, but black holes do have angular momentum, and there is a dimensionless number relating that to their mass (or equivalently radius) which can be loosely thought of as the fraction of light-speed at which the event horizon is moving. This paper measures a value of about 0.44 for that parameter for the black hole in the centre of the milky way.
At the other extreme, this stackexchange answer suggests that the electrons in a hydrogen atom rotate about the nucleus at something a bit less than $0.01c$. This scales up linearly with the charge on the nucleus, so the inner electrons in a lead or uranium atom are in some sense rotating at close to light-speed.
Finally, the particles in a large particle accelerator, rotate around the centre of the accelerator at velocities very close to that of light.