You should dispute the overpayment using a form TC846, stating you have made several attempts to ensure the amounts paid to you are correct and that you were informed by their staff that the amounts were correct. HMRC will suspend recovery of the overpayment until the dispute process has ended. This will at the very least give you a little more time to sort things out.
This link
http://www.hmrc.gov.uk/leaflets/cop26.pdf will take you to the HMRC guidance regarding overpayments including what you should do when there is an overpayment and what your responsibilities are.
HMRC have a system whereby every time you contect them or they contact you or they send a letter or do anything to your case, notes are recorded on your record explaining what has happened. They should be able to check these and see you have made repeated attempts to ensure the amount paid to you is correct.
This does not ensure that you will win and have the amount written off. If you have filled in the tax credit claim form incorrectly, or have not checked the renewals declerations before sending them back into HMRC each year, it is likely they will say you are at fault for providing incorrect information as the amount paid to you was based off of what you have provided to them.
If you do lose, HMRC have several payment options available, including reducing any current tax credits you are receiving or allowing you to pay over a 12 month period. Even so just under £1,000 a month would cause hardship for most people, so I would suggest you contact them do come to some kind of arrangement if you have to pay them back.
Hope this helps.
overpaid tax credit £11000 
