Can a car dealer take your car if you haven't pay your full deposit but they told the finance bank I did pay in full?

Your contract is with you and the car dealership. If you break your contract with them, why wouldn't they? They are trusting you to keep your Word. If you choose to screw them by not paying it, you should pay the price.

Until the note is paid in full, the repo option is always on the table if you show yourself to be a bad risk in danger of default. If you tell them you will default ahead of the time when that will happen, they have the right to repo your car on a pre-emptive basis.

Remember, they are at risk from you. They will not repo their car from you unless you prove to be not worth their trust, as evidenced by not keeping your Word by withholding their money from them.