The direct-only lender, which is part of HSBC, has created an appointment system but customers are being forced to wait at least two weeks.
The most popular deals include a lifetime tracker mortgage at 1.79%, 2.29% and 3.49% above base rate, available at 65%, 75% and 85% LTVs respectively.
But a spokesperson says no one will miss out on a mortgage.
She adds: “The system we have set up means everyone will get an appointment over time. We are bringing people in to deal with the mortgages we have got in.
“It’s not that we can not handle it but it has been a very popular deal. I think it is the flat rate that has attracted people.”
Last month the lender reported a record weekend after introducing the £99 flat rate fee.
In early July Richard Tolchard, senior product manager for first direct, said: “We are delighted with the response we’ve had to our new £99 fee offer.
“It’s a challenge for consumers looking at all the hundreds of mortgages available and we know we offer some of the cheapest mortgages in the market. Customers clearly like the clarity of our low rate low fee deals.”