LocationIt can be found in the first part of the game, when you are on Earth at the Ramsey Recruitment Center in training. By turning to the right once you are outside on the main street, climb on top of a nearby ramp, where you will find the basketball. It is able to be picked up and it will remain in inventory until it is dropped.
Keep the basketball until you are on the Recreation Deck of the Von Braun. Enter the Athletics Sector and once you find the basketball court, dropping it through the basketball hoop will cause a secret message to play. Afterward, there is no further use for it. Either leave it on the basketball court, or recycle it for 1 nanite.