It really looks like something is manipulating the page DOM. The drop-down list as well as the button to submit it are missing, and they aren't loaded in via JavaScript (so shouldn't be an issue with AJAX or JavaScript not loading).
Are you able to try a different browser/device to see if it's something specific to that device? Maybe an ad blocker (they do a lot of DOM manipulation and tend to cause a lot of problems)?
It's possible something third-party in WordPress may be doing it as well. I've seen plugins that are supposed to block ads in the admin interface do it incorrectly, and also block things that it shouldn't be blocking.