There's no such feature built-in in PDF files, no. However, it is possible to achieve something similar using a custom-made script. It will not act exactly like a regular drop-down, though, in the sense that it won't close automatically if you click elsewhere on the page. You will have to make a selection or click a Cancel button to close it.
If that's OK with you and you're interested in hiring someone to develop it for you then you can contact me privately via [try6767 at gmail.com] to discuss it further.