The car has to be fit for purpose - which means that the buyer can expect to get a car that performs for him much like any other similar car with a mileage and condition that merits paying about this much for it.
The seller must also give as true and accurate a description of the car as possible - meaning that you can't tell porkies about its condition in the hope of ramping up the price
My understanding is that if the car breaks down because of normal wear & tear or a routine 'problem' - such as a failure of some component that A - is problematic on this type of vehicle or B - its not hugely surprising that it wears out at this mileage given the car's overall condition, then this is the buyer's problem - caveat emptor and all that.
What would be the seller's problem is where a major fault on the car was not reported or (worse) covered up - for example, structural rust.
In the event of something fundamental being wrong with the car, even where its a private sale, the buyer can look for money back or recompense of some type if the car falls seriously short of its description or price - even if the seller didn't know about the fault - because its reasonable to expect that something priced like its in good condition actually is in relatively good condition.
In the case of the private sale, I believe its the buyer's job to demonstrate all of this. Not the easiest for them to do, but it is possible.
Sounds too unclear in this case what the issue is. If the buyer continues to complain, then it might be an idea to ask him to conduct all further contact in writing (to keep everyone/thing right & clear), to specify exactly what the problem is and reserve the right to get your own inspection of the car done to verify any claims made about its condition.