Can nuclei ever "liquify"? I've been reading about degenerate matter and how extreme temperatures and pressures can impact the way it organizes itself. Plasma is a type of matter where the energy of the matter allows electrons to flow freely through it in a sort of soup. 
Could matter be pushed into a state where the nuclei drift apart and a "fluid" of protons, neutrons, and electrons is all that's left, with perhaps only small atoms left? If so, what would be required to form this type of matter?
Edit: to clarify, I'm not looking for quark-gluon plasma. I'm specifically talking about a state in which the nucleus ceases to be but the protons and neutrons remain intact.
 A: This is exactly what happens to nuclei in a neutron star as densities exceed about $10^{17}$ kg/m$^3$; the bulk of a neutron star (by mass) is probably in this phase. What is required is a high density, which creates high particle Fermi energies, allowing extreme levels of neutronisation. As a result of the high Fermi energies you also need the stuff to be gravitationally confined in a stellar-sized lump, because the material has incredibly high internal energy density.
The nuclei first become extremely large and neutron rich, then they become surrounded by a degenerate neutron fluid above densities of $4\times 10^{14}$ kg/m$^3$. At about $3\times 10^{16}$ kg/m$^3$ the nuclei become unstable to fission, but because they are so closely packed, rather than break into smaller, isolated nuclei, it is energetically more favourable to form strings and sheets of nuclear matter known as nuclear pasta. The exact shapes are determined by competition between surface energies and Coulomb repulsion due to the remaining nuclear protons.
Finally, the binding energy of these constructions is insufficiently large, the protons start to drip out of the nuclear matter and it becomes energetically favourable for it all to dissolve into a fluid of mainly neutrons, with a small (1%) fraction of degenerate protons and electrons in "beta equilibrium".
The schematic below is taken from a well-known review by Chamel & Haensel (2008) and shows the progression with density moving into the neutron star interior.

A: Yes, such a state is called a quark-gluon plasma. The atomic nuclei, and even the individual protons and neutrons that comprise it, dissolve and the individual constituent quarks run around only weakly interacting with each other. It occurs at extremely high temperatures and pressures. It is believed to have been observed experimentally at high-energy accelerators, but the situation is not conclusive.
In response to your edit: I'm not sure, but I think there's only one actual phase transition, at which both the nuclei and the nucleons dissolve simultaneously. But there's probably an intermediate temperature range just below the Hagedorn temperature in which the nucleons are much more tightly bound than the nuclei, which would qualitatively look like such a state on the intermediate length scales between the binding radii.
