P45

My 16 yr old daughter had a part time job in a local cafe since middle of May and has just been told they don't require her anymore as they don't think she will be able to keep up with the washing up during the busy summer month. She isn't too fussed about not working there anymore.
I have asked for her P45 and there response was "I've no P45 as she wasn't on the books with the short amount of hours completed"
Don't they legally have to give her a P45? They where paying her directly into her bank account. Want to get facts correct before I go back to them

You are correct. The must, by law, issue a P45. The fact they they say she "wasn't on the books" sounds like they've not been processed her wages through a PAYE system, and so have broken the law already anyway.

A couple of quotes from .gov.uk to confirm the above:

"By law your employer must give you a P45 - ask them for one." (source)
"You must give your employee a P45 when they leave." (source)

My guess is they will argue that she wasn't actually an employee, but was a "casual worker". Regardless of how they classify the arrangement, she will have been considered to be an employee by HMRC. Hence a P45 must be issued.