If a massive sphere spins faster and faster, will a black hole appear?

If we make an object moving faster and faster in 1 direction no black hole will not be produced. We can always find a frame in which the mass is at rest. Even though we have put a lot of work into it, the rest mass will not increase. Only the relative (relativistic) mass increases. This can be used to make a black hole appear if we let the mass meet with another (equal) mass in a frame that is at rest wrt to the frame of the speeding mass. Meaning that the increase in mass is indeed relative.
But what happens if we make a massive object rotate with increasing speed? We put more and more energy in it so the relative mass increases. The frame in which the mass is at rest is not an inertial frame, as in the linear case. It is a co-rotating frame in which a gravity field is present but the mass is at rest. So you would expect that in this case a black hole is formed without the need to let it meet another mass like in the linear case (the increase in mass is not relative?).
Because the gravitational field of the rotating mass is determined by the (rest)mass and momentum the field will be different from a static mass in the sense that frame-dragging is present. Will this lead to the formation of a black hole? Is the Kerr metric important in this case? I mean, this is the metric of a rotating black hole, of which I want to know if it appears in the first place.
Will the only effect of the spinning be that space is super-dragged? In other words, will there be no change in the radial part of the metric but only in the radial part? Will the radial part show time dependence?

So, in short:
Let’s assume that the object is an incompressible sphere, to avoid deformation. Let’s ignore the paradoxes of length contraction (or can’t we?). How will the spacetime around the sphere evolve if it rotates faster and faster?

One last thing. Suppose we look at the sphere at an instant. It is composed of infinitesimal pieces of mass that all have a linear velocity. Each piece of mass is contracted (that is, the space it occupies) due to length contraction, which will give problems. How can a sphere contract in an angular direction? Will its volume become smaller? But aside from that, can’t we say that because linear moving masses do not result in a black hole, a rotating sphere will not either? A linear moving mass will only give rise to frame dragging in the direction of motion, so is this not the case here too? Or can you say that because the rotational kinetic energy of the sphere increases (which is also the case in linear motion, though there no black hole arises because the linear kinetic energy does not stay in the same volume) a black hole must see the daylight? So will it only be frame-dragging or will a hole be there? It must be the last, because there will be kinetic energy present in a "stationary" volume (unlike the linear case) that approaches infinity. Can we say that the sphere, due to special relativistic effects, shrinks to a sphere of zero radius (though mechanical incompressible or inexpandable)? Which results in a black hole? On the other hand, will the centrifugal force not be able to prevent the hole from forming? Suppose a neutron star, as mentioned by @Andrew, was rotating that fast that the gravity force holding it together would be comparable to the centrifugal force (both forces are proportional to $frac{1}{r}$). The star will never become a hole in this case (the question of how the star got in that state if a different one though). It’s complicated indeed…