Most of the posts so far seem to have dealt with the termination of the contract, rather than the overpayment. With certain exceptions (which don't seem to be relevant here), 'unfair dismissal' only applies to people who've been employed for at least 12 months. Apart from the limited exceptions I've referred to, an employer is free to dismiss any employee, for any reason (or even in the complete absence of a reason), within the first year of employment.
With regard to the overpayment (assuming it to be genuine), your daughter should make it clear that she's not simply refusing to repay the money. Deliberately 'hanging onto' money paid in error is a criminal offence under the provisions of the Theft Act 1968. As long as your daughter makes it clear that she can't repay the money immediately (rather than simply refusing to do so) the matter falls solely within civil law, rather than criminal law.
That means that she's in exactly the same position as the hundreds of thousands of people who can't meet their credit card payments. She should offer to pay a nominal amount (of, say, a pound or two) each week. The former employer can choose to either accept the offer or to pursue the matter through the courts. If they do so, all that will happen is that a County Court Judgement will be granted, requiring your daughter to repay the money at an affordable level (which will probably still only be a pound or two each week anyway).
Chris