Chemistry - Are there ionic solids that conduct electricity?

Solution 1:

Yes.

Another example is VO2. Over 67 °C its crystal structure changes to rutile with a band gap of ~0.1 eV. Dopants of other substances can reduce the temperature at which it becomes a conductor.

Solution 2:

Yes, there are.

Ionic conductivity greatly varies between compounds. For some of them is much higher then for others. They are called fast ion conductors, or superionic conductors.

Well known example is silver iodide. While it may be called a "salt", it's probably more covalent then ionic, that is, before it's heated to 146 °C, when its structure changes and Ag ions are "set loose" to conduct electricity. In result conductance rises a few thousand times.

If rubidium iodide is added to $\ce{AgI}$ we can get rubidium silver iodide which is even better, and is sometimes classified as one of "advanced" superionic conductors.

Many substances can be superionic in right circumstances, even water ice can, if temperature and pressure are high enough.