The electrolyte as an ionic conductor causes most of the ohmic (resistive) losses in the capacitor.
Alternatively, ions support electric current in ionic conductors.
Calcium bromide is an "ionic conductor," and so conducts electricity only at relatively high temperatures.
With the addition of cations like Na can be made into an ionic conductor.
This material is considered a mixed ionic electronic conductor with transference number close to 1.
So you were correct to notice that mercury or molten sodium are metals and electron-conductors, rather than ionic conductors.
There are ionic conductors that are solid.
But iconically conductive solids and liquids share this correlation: ionic conductors are not metal-colored.
So there will be no ionic conductors, liquid or solid.
It is also clear from the picture why glass can often be good ionic conductors.