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.