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Patients are waiting up to a month to see their doctor.
A shortage of routine appointments at Newton Place Surgery has pushed waiting times up to between three and four weeks.
The GP practice, which has 18,000 patients, is under strain because this year's summer holiday season has coincided with a number of staff being on sick leave, and difficulties finding a locum doctor or nurse practitioner to cover.
Despite doctors working extra shifts to cover the shortfall, the shortage is having a knock-on impact on waits at the surgery's urgent care clinic, which can have over 100 patients daily.
Patients seeking a same-day appointment are currently waiting for up to an hour and a half.
Operations Manager Claire Boraston said: "Normally we would aim to have our waits [for routine appointments] at two weeks. That's the maximum we would want it to be.
"We don't see this as long term and we hope the situation will get better in the next month. This is a short term blip in our system, but we want to be open with our patients about it."
Ms Boraston is asking patients to do their part by ensuring any appointments which are no longer needed are cancelled in good time so they can be allocated to others.
A total of 353 booked appointments were wasted last month after patients failed to cancel - at ten minutes per appointment, the equivalent of a week of a doctor's time.
"What we do need is to make sure that patients cancel their appointments if they can't attend. And if someone feels that it's urgent, they should come to our urgent care clinic," she continued.
Patients are also being encouraged to go to their local pharmacy if experiencing minor ailments such as coughs, colds, aches and pains.
The surgery, in Newton Road, recently submitted plans for a £1.8m extension which would see its premises expand by 75% of its current size.
With its patient list growing on average by 600 a year, doctors say this extra room is much-needed, particularly with more than 1,300 houses due to be built in and around Faversham in the coming years.
The practice's urgent care clinic continues to run as normal, from Monday to Friday between 8-10.30am and 3-6pm.