Honestly, I think it would be best to have them select a country and then just enter their address in a multi-line text box. That will include their city, postal code, and street address. It would take way too much space to list every city on the planet for them to select from a dropdown and the functionality would be insanely slow.
That being said, there may be API’s through Google Maps or something similar that provide clickable and usable selections for cities as they type something in. For example:
User typing: Chic…
Suggestion pops up below the text box: Chicago, USA
Just a thought. Let me know if I didn’t understand your question correctly. Best of luck.