"Would you give me your car?"
Translation:Dałbyś mi twój samochód?
The particle by should never be put in accented positions (e.g. at the very beginning of a sentence).
That actually sounds like "So you (can) give me your car", or something like that. As if "Byś" was interchangeable with "Abyś" or the most common "Żebyś". So it would be like "I am helping you out because I hope that as a reward you can give me your car for the weekend".
Of course, I haven't thought of that. However, byś with this meaning doesn't really work in a standalone sentence without a main clause, right?
Well, it may just be an answer to a question like "Why are you helping me?". You can say that "Robię to," is an implied main clause ;)