If this is an issue concerning unemployment insurance eligibility, I would not try to use "falsifying of payroll records" or submitting a false request for pay. Because if she was on the clock, and she HAD been told not to do the job by her supervisor, even so, she would not have been taken off the clock. If he told her to clock out and leave instead of doing the job, and she chose to stay there, then it would still be more insubordination than time sheet fraud.
Frankly, I do not think, if this is the first offense, and if there are no prior documented disciplinary offenses of this nature, that this "rises to the level of" a firing for misconduct type offense.
I think the person would probably be approved for U.I., because she will maintain that this was a misunderstanding, that she had been told not to do the job. And your word and hers will have equal weight. Even if the supervisor testified he did not say it, she would say she misunderstood what he was telling her, that she had no idea her job was in jeopardy, and that she had no prior warnings or write ups about this type of behavior.
So give her one. If your business has a prescribed chain of discipline, eg oral, written, suspension, termination, begin her progressive discipline. Even if she denies she did it, you can give her a warning about it. Keep these records. Your goal, of course is to make sure the behavior is not repeated, more so than to punish.
If the behavior does continue, then you will have a trail of warnings and attempted correctons of the behavior, and can terminate her without repercussions to you. It will be, of course, her choice to re-offend or not.