Technically, any real solid above 0 K also flows really slowly. In a crystal lattice, you get flaws - vacancies, dislocations and grain boundaries. At the right-hand tail of the energy distribution of the atoms in the solid, some will occasionally have enough energy to jump from a lattice site to a vacancy. This allows the solid to gradually change shape.
For high temperature applications, the slow flow of solids over time can become a problem, and is called creep. Gas turbine blades can elongate over their life by creep. Pure tungsten lightbulb filaments can creep under their own weight, which is one reason why the filaments also contain thorium oxide dispersions. Lead, which is a crystalline solid, creeps measureably at room temperature.
Now, as you lower the temperature, the creep rate becomes so small it’s immeasurable. But theoretically it’s still happening, even if it’s only a few dozen atoms per year. Same with the “flow” of glass. Extrapolating viscosity vs. temperature, you can claim that glass is flowing at room temperature if you want. But you’re not exactly going to be plugging that viscosity into a fluid dynamics equation and pumping room-temperature glass down a pipe!
I’ve only encountered the flow of solid rock in a geological sense, where it happens under very high hydrostatic pressure. In terms of flow, a liquid is a substance of fixed volume with zero shear strength, and a solid is a substance of fixed volume with non-zero shear strength. If you apply a pressure to a solid hundreds of times greater than its shear strength, the difference between zero shear strength and comparatively tiny shear strength becomes neglible and the solid will obey the hydrodynamic equations for liquids quite happily.
Bit of a sidetrack, but you may be family with the HEAT-type anti-tank round, or “shaped charge”. This uses an explosive block with a conical hollow cavity, lined with steel or copper. The explosion squeezes the liner like a lemon pip between the fingers, firing a “jet” that punches through the tank armour like a screwdriver through styrofoam.
What is interesting about the HEAT round is that the jet isn’t liquid metal - flash X-ray crystallography shows that it’s solid crystalline copper or steel, although under such huge pressure that it flows exactly as if it were liquid. Also interesting is that glass can function nicely as a cavity liner, a hollow glass cone happily squishing into a glass “carrot”.