I had a chicken who caught her toenail jumping over a chainlink fence. It did not grow back. I still suspect that your chicken may have had some slight frostbite, but it looks like she will be fine. I have had chickens lose a tip of the toe from frostbite, when temps reach near zero.
How cold has it been there at night? If she gets her feet wet, in freezing temps, she will get more frostbite. A smash injury to the toe can look similar, but if she is outside, it’s frostbite. I would not put anything on it if she stays outside. It will self amputate in a month or so. Here is a...
Do you not have a chicken coop for her? It looks like her toe has been frostbitten. The tip will eventually fall off. What is the dark stuff on the right foot?