You could progressively enhance a select by replacing it entirely with a label and style the label to make it look like a dropdown menu, then attach an event handler on the click and possibly keydown event to open up sibling div that contains spans which contain value attributes ( or hold the data in the js but it's easier this way ) and then attach separate click event handlers to the spans. you'll need to attach a click handler to the document and if the event object's target isnt inside of a generated dropdown, close it, and you'll probably need some other logic.
This will probably be non-accessible at first, you could try to make it more accessible by attaching keyboard event handlers but it's practical.