While in a parking lot moving forward at VERY slow rate speed (about 1 mph) the car suddenly accelerated on its own.

I had just slowly turned within a parking lot in preparation to backing into a parking space. I had just removed my foot from the accelerator and the car was almost at a stop and I was about to brake and put the car into reverse when it suddenly and rapidly accelerated on its own. My foot was not touching the accelerator at the time (it was over, but not yet touching, the brake). The car felt as if someone had suddenly slammed on the accelerator, but my foot was not on the accelerator at the time.

Luckily my foot was over the brake when the rapid unexpected acceleration occurred and also luckily there were no cars in front of me. I was able to immediately slam on the brake so that the car did not hit anything. Before I was able to brake, the car quickly lurched forward at a surprisingly rapid rate of speed for a short distance. Quick application of the brakes stopped the sudden and rapid forward movement of the car.

I did not notice any warning lights or messages on the car.

This could have been a very dangerous situation. If someone had been walking in front of the car, they would have been hit and potentially seriously injured. If there had been cars parked in front of the space I was about to back into, the car would have hit them and caused damage. I was very lucky to avoid an accident.