Unless new technology is just shockingly and suddenly better now and some advanced system can, but I seriously doubt it. Water is one of the hardest things for even expert human sound techs with computers to deal with, let alone computer-based real-time cancellation systems.
The reason is that the rate of change of the sounds is very high - higher order derivatives included. Most human- and machine-made sounds have a steady rise and fall rate after the initial uptake, and almost always a constant second derivative (the rate of getting louder/quieter is increasing/decreasing steadily), so the system can track that and predict the next several waves pretty accurately. Secondly, the sound of running water is made of innumerable components, unlike the one or two dozen that make ambient indoor or outdoor sounds. Water also has the serious problem that there is no point when the noise is low.
You have a rapidly changing sound (in first and higher-order derivatives), a high minimum level of sound (ongoing sound), made of innumerable components. Good luck predicting anything even at the processing rate of two or three sound waves.
You may reply, as my friend did, that ambient noise cancellation can cancel even something bumping into the wall somewhat loudly, so it can react very quickly. First of all even bumping into the wall starts at a very low level and rises quickly, but there is an initial warning at very low volume and a convex rising in its volume. The system can track and predict several waves into the future. Also, considering the rate at which sounds are being processed, the system has to deal with this sound almost by itself, not the equivalent of 100,000 (or millions) of other bumps per minute like water. So it can recognize it very early.