Wireless deskphones do exist. I have used Grandstream phones before and they're not the best. The build quality isn't very nice and the software can be a little buggy. I've even had one that ran Android (the one I linked probably does, too).

I don't know how reliable the call quality on a phone like that would be, but we have someone in the office that I gave a UniData WPU-7800. It's a handheld WiFi IP phone. He uses that thing all day walking between the office and the warehouse, moving between access points, and there are never any issues. Wireless might be a viable option for some of your extensions.

Whatever you do, I would suggest you standardize on one phone model. So, try to find something that ticks all the boxes for every situation. Also keep in mind that you can mess around with powerline networking or perhaps using a separate WiFi client bridge. I have a Polycom VVX 500 set up on powerline at home for when I work remotely. I've been on calls for hours at a time and never had any hiccups.