Bill is right, a BB gun won't do it !  A decent air rifle even at Walmart will go $100 to $150.  I paid about $150, several years ago, fo ra Ruger Airhawk
  with wooden stock.
    Although mine is a .177, I have to admit that the .22 though slower, delivers more "punch" on the target.
 
   Frankly, I almost never use the Airhawk anymore. When I am troubled with a bothersome critter, I just use a .22 LR, with a Colibri or CCI Quiet.
 One of the serious air rifles require maintenance, perhaps more that a regular .22 bolt of single shot. With a colibri, use a shorter barrel, since with a 
   longer barrel..24" on up.  A longer barrel raises the possibility of a bullet lodging in the barrel.
     A Rascal or Cricket sounds good to me, depending upon your situation.  Then again, the Rascal is a self-cocker with accu-trigger.