Yes, running water can freeze, although it requires specific conditions.
Even though moving water, like that in a river or stream, constantly mixes, preventing uniform cooling, it can still freeze if the temperature stays below freezing for an extended period. The water becomes supercooled (below 32 degrees Fahrenheit but still liquid), and frazil ice starts to form. This ice can then accumulate and eventually freeze the entire body of running water.
Think of waterfalls in winter. They can freeze, creating spectacular ice formations, demonstrating that even rapidly moving water isn't immune to freezing temperatures.
