Can all matter be classified as solid liquid or gas?

No. While most matter on Earth is in one of those three states, not all of it is. A gas that ionized (electrons have broken free of atoms or molecules) enters a fourth state of matter called plasma. Plasma can be found in electrical arcs such as lightning and where substances are heated to extreme temperatures. The sun is plasma.

At high temperatures and pressures some substances that are normally liquids or gasses can become supercritical fluids, a state that is intermediate between liquid and gas.

At temperatures near absolute zero some materials can from a Bose-Einstein condensate, in which atoms stop acting as individual particles and start acting as a uniform wave.

The cores of many dead stars form white dwarfs, which are are composed of electron-degenerate matter, a state thousands of times denser than anything found on Earth.

The cores of massive stars may form neutron stars, which are composed of neutron-degenerate matter, which is millions of times denser than electron-degenerate matter.