Chop down the banana tree with an axe, leaving about a 2- to 3-foot stump. Discard the rest of the tree, and pick the ripe bananas, if you wish.
Chop an "X" into the top of the banana tree's stump with an axe, then pour 1 to 2 ounces of lamp oil or kerosene into the "X." The tree stump will absorb the oil or kerosene, and it will begin to kill the tree stump and weaken the roots within one to two weeks.
Wrap a tarp around the stump tightly, and secure the tarp with with a rope. This will rob the banana tree of rainwater and sunshine, preventing it from regenerating while the lamp oil or kerosene works.
Dig a circle around the banana tree about 2 feet from the trunk after seven to 10 days have passed. Dig until the root ball is exposed, and then dig under the root ball. Pry the deadening banana tree up with the shovel.
Pull out any existing plant corms, or underground stems that banana trees use to reproduce, to keep new trees from growing. Re-fill the hole.