An automatic response should include the date when you left, a reason (vacation, sick leave, a conference, etc) and, if possible, the date when you plan to come back. Thank your client for their email, promise to get back to it as soon as possible and apologize for the inconvenience. Also, it should be obvious that you’re out of office from the first sentence.
When one of my colleagues went on vacation, he sent an out-of-office message that was both clever and smart. First, he sent the recipient on an imaginary scavenger hunt to “the highest peak of the tallest mountain.” He used humorous absurdity to make it clear that he would not be checking email while he was away.

I wouldn’t be surprised if she’s had experiences in the past with people not getting an immediate answer then upping the urgency–we’ve had letters about such coworkers here. There’s an email, then a followup email, then a chat message, then a phone call, then they walk over, all within ten minutes of the initial email.
I can see how they would annoy given her character at work though. There is something especially cruel about advocating for your boundaries while disrespecting other peoples.