WHEW! I'm ahead of the "but can you eat them?" question this time...lol
In my opinion, 17HMR is basically made for gophers. I think the way ammo costs are going, you need a good reason NOT to pick .223 over 17HMR/22WMR these days..but the magnum rimfires do have their place. .223 is still more expensive but brings allot to the table for the money...and the same is becoming less true with the big rimfire calibers. In my opinion. I've never shot a gopher with 22WMR, but I've shot a bucket-load with 22LR (my preferred caliber) and have introduced hundreds of gophers to the charms of 17HMR. I spent a summer shooting groundhogs here in Ontario with 22WMR when I was getting into that. For a far bigger animal than a gopher, I'd say 22WMR is the better choice..but beyond 100-125 yards, I was mindful of the accuracy limitations.
I use a CZ452 Varmint, 22LR, as my main gopher rifle but if I were settting-up another, it would be a Tikka T1x MTR. I think the rifle/caliber choice depends on allot of factors. I travel to shoot gophers and try to drop myself into areas with lots of targets. 22LR makes the most sense for me, but we have a 17HMR as a back-up. Long shots, going for multiples, really need to see their geometry change=17HMR is more fun than you can shake a stick at. With really big numbers, property owners are hoping for big results so challenging yourself on the long shots is fun...but volume is kind of the wish/expectation. You don't need 17HMR to be popping these things inside of 50 yards. Or 40. Or 30. Still fun, but so too are 22LR HV HPs.
I once spent some time in a gopher patch with a buddy's SAKO VArmint in .223, with 55gr. V-Max hand loads and some amazing scope I don't remember. Hits were remarkable. If you've done it, you know.
Good luck, and post pics!