Break Free CLP would be a good bet overall. It's what I've used over the years for rifled barrel pellet rifles spring loaded (break barrel or lever) action. If it's CO² then (oddly enough) some air tool oil might help keep the gaskets, o-rings and things pliable and in good working order on the insides for the action. On the outside and for the barrel, you'll want the CLP treatment of some sort.
Daisy BB guns or pellet rifles are a bit different in ways than what one normally considers "firearms". I had just recently schooled a co-worker about choosing a scope for pellet rifle compared to rimfire or centerfire scopes. Truth be known, you may find more metal content in air rifles these days than other modern firearms. The only thing you don't have to be concerned about with the air rifles/pistols is the carbon and powder build up/fouling.
Some of the current production air rifles get close to what a 22LR can do. Those might be more expensive than a rifle chambered for 22LR!
We can't technically consider them firearms by definition, but they are closely related in caring for and maintenance.