There are some air rifles that are powerful enough for deer hunting. They can fire lead bullets up to at least .50 caliber. I did find a brief mention of a .89 caliber but it seemed to be European.
The rifles are very expensive. You will also need a pump to fill the high pressure air tank. The hand pumps don't cost much but the electric compressors are very expensive. Ammunition is very cheap. Buy a bullet mold, a heat source, some lead, a ladle and you have ammunition that is CHEAP.
Look up PCP air rifles. It's amazing how many different makes there are.