I fear that my Samsung American style fridge freezer is on it's way out - the fridge section keeps freezing up (as in iceberg on the back wall originating from the cooling unit. It's manageable with a defrost every 6-10 months and I've now removed the shelf closest to the cooling unit, which has improved it a lot
It may be a failed heater, or relay feeding the heater, or the fridge temperature sensor or some other component that can be replaced.
Also, when defrosting a frost-free fridge, make sure that you let it get properly warm. The actual cooling coils are hidden, and the only thing that carries cold to the food compartment is air blown by fans. If the visible ice has gone, there may still be ice blocking the cooling coils. The heating cycle of frost-free fridges may only work correctly for removing small amounts of ice from the cooling coils, and if something causes a complete blockage with ice, the normal running of the fridge won't remove it. Also the hidden cooling coils are deliberately arranged so that heat from them cannot escape without the fans running, so when defrosting, the coils will take far longer to melt than the visible ice that will gain heat from the room as soon as you leave the doors open.
I fixed one that was freezing up where a relay on the control board had welded contacts, resulting in the fridge needing defrosting regularly. I never saw the fridge freezer, as it was a colleague's, and he brought the control board to me to fix. Apparently there are other common faults in the control boards, and my colleague had bought a set of replacement capacitors for it. I changed them as well, but the ones I took out were fine. The relay was welded in on position and never operated.
The cooling system faults, of the type Carnardly has suffered, are very rare and should be just as likely on simple fridges as on complicated ones, because the cooling systems are virtually identical, don't have much in the way of new technology, and have proved to just work for decades in most cases.
Faults with freezing up are confined to frost-free fridges and freezers, and there are lots of bit for the defrost systems that can go wrong, but can be fixed.