If the person at rest flips a coin a certain height in the air and they time it as taking 1 second to return to their hand then the person in the car will be able to perform the same experiment (assuming you can exactly reproduce the coin flip to the same height) and record the same result in their frame of reference (1 second).
The person at rest will observe the car travelling at 99.9% the speed of light to be effected by time dilation, e.g. the time in the car will appear to be travelling slower relative to their own frame of reference (that of rest). As such the person at rest will observe the flipped coin in the car to take longer than 1 second (as measured from the frame of reference at rest).
Note that I've avoided them flipping at the same time as they will be at different places when they start and end the flip and you can't tell if the events occur at the same time or not. Hopefully though the scenario above answers your question, the person at rest will observe the coin flip in the car to take longer to land than what they would expect if they had performed it in their frame of reference and they explain this by saying that time dilation effects the frame of reference moving close to the speed of light.
I'm ignoring the big planet part and just assuming a uniform gravity throughout both frames of reference.
See XKCD's Relativistic Baseball for an idea of what happens if your car travels at relativistic speeds in an atmosphere.
Flipping a coin is quite random and hard to reproduce, you and the person at rest will need to practice this a LOT to get it exactly reproducible each time.