Frozen Mud is called Permafrost.
Permafrost is a type of frozen ground that stays at or below 0° Celsius (32° Fahrenheit) for at least two years.
Permafrost does not have to contain water or ice.
As long as the temperature of the ground stays below freezing, it is still considered frozen ground, even if it is completely dry.
Mud may also just freeze when it's really cold out and then thaw out once the weather warms back up again.
In that case it's just called frozen mud.
If the temperature is below freezing or at least freezing and there's mud then that mud will freeze over and be hard until it thaws out when the temperature gets above freezing.
But if the temperature is still freezing then the mud will also remain freezing until it warms up enough.