A high powered break air rifle will weigh 8+ lbs and practical hunting will be limited to 35-45 metres. Ammunition will very light though and you could easily carry 1000s of pellets.

But if your requirement was for a backup then a Ruger Mark II Stainless Slab side Competition Target Model 22LR pistol might be worth looking into.

Edit

How about one of these

http://www.crickett.com/crickett_hunterpistol.php



Cricket Hunter single shot pistol .22LR

You should also be able to get a full replacement stock for more control which would take the weight of the rifle to around the 3lb mark.