Pedro de la Torre said:
Does a degeneracy pressure for protons exist?
Yes, theoretically speaking. But practically speaking, it never gets observed, because, as
@Nugatory said, at the densities where it would become significant, protons and electrons are forced together to form neutrons.
In other words, theoretically, if you could assemble, say, ##10^{60}## protons into a compact object like a neutron star, without including any electrons, then yes, the structure of that object would be significantly affected by proton degeneracy pressure. But assembling such an object is impossible in any practical sense, because the protons would repel each other electrically and you would never be able to push them together into a compact enough object in the first place. Neutron stars are able to form because the objects they form from, ordinary stars that go supernova, are electrically neutral--they contain equal numbers of protons and electrons. That is what allows them to get compressed to the point where degeneracy pressure starts to matter--but, as above, by the time proton degeneracy pressure would start to matter, the protons and electrons have all been forced together to form neutrons.
(I have not done the math, but I am skeptical that such a compact "proton star" would be stable even if you could assemble it; I don't think its gravity would be sufficient to hold it together for any long period of time against the combined effects of proton degeneracy pressure and the electrical repulsion between the protons.)