We know that ice is already the frozen (solid) form of water. The question is more like: Can this frozen form freeze further? Or can it become more solid? (for example, by exposing to colder temperatures and/or a higher pressure).
Water can partially freeze but we can think of a solid frozen state. It is said that virtually all ice in the biosphere is ice Ih (hexagonal crystal form of ordinary ice), but there is also Ice Ic (metastable cubic crystalline) that is present in upper atmosphere in very low temperatures (between −140 and −50 °C). [There are also different forms of ice created in labs.]1
Furthermore, scientists predict that there is an out-of-this-world kind of ice that is present at extremely high pressures nonexistent on Earth but in the range of the pressure regimes on Uranus and Neptune (planets whose major components include ice). At this pressure, oxygen-hydrogen-oxygen bonds get squeezed together, forming new shapes. The new ices eventually become metals, but not as quickly as others had previously thought.2
Static crystal structure of a phase of ice at a pressure of 2 terapascals.2
In the end, can we say that different crystalline forms have different solid states? Can ice freeze further by transforming into a different crystalline form?
Further clarification: "More solid" and "freeze further" might not sound scientific but I wanted to ask in that way for simplification. I meant changes in material properties by saying "more solid". For example,change in hardness, molecules getting closer, stronger bonds etc. [It is even said that ice can become metal at very high pressures.]
Also, when ice is changing solid phases, what is happening there? I thought it is transforming into a harder structure in some phases almost like "freezing further"? I thought this might happen when ice changes into another solid phase. That's why I ask if ice can freeze or not.
Note: There is also fractional freezing but it is related to solutions mainly, for example that contains water and alcohol. It is used in freeze distillation processes.