I know that white dwarfs are composed primarily of carbon but are supported by electron degeneracy pressure. Is the electron degeneracy pressure produced by carbon atoms, or is the core of the white dwarf made of electron degenerate Fermi gas while the outer part is atomic carbon? I am trying to reconcile the fact that white dwarfs are made of both carbon and electron degenerate matter. Can carbon be electron degenerate?
Most white dwarfs consist of fully ionised carbon and oxygen atoms$^1$. Each carbon nucleus must be accompanied by 6 electrons and each oxygen nucleus by 8 electrons. The electrons are essentially a free Fermi gas around the nuclei. All of this can be referred to as "electron-degenerate matter".
It is however the free electrons that are degenerate and which exert the degeneracy pressure that supports a white dwarf. The carbon and oxygen nuclei are far too massive to become a degenerate gas (and they are bosons too!).
$^1$ We are talking here about white dwarf interiors, not their atmospheric layers which can contain hydrogen, helium and other metals and are not degenerate.