The Battle Master's Pushing Attack can push the target up to 15 ft away from the user if the target fails their save. Ordinarily, this would mean the target is driven back 15 feet away on the ground, as with the Shove attack.
However, I was thinking of a scenario that involved two characters falling: one Battle Master and his target.
If the BM strikes his target, hits, and uses Pushing Attack, he would reasonably be able to drive the target 15 ft in any direction, because there's no floor that anchors our mind to a horizontal surface, unlike the regular "shove" attack. In fact, pushing someone directly perpendicular to the floor in this situation is much harder than pushing someone sideways but with some up/down elevation.
Taken to the extreme, in this case, the BM could even shove the target upwards. If they hadn't fallen more than 15 ft yet, the target could "return to safety" by being thrown back up to the platform they fell from. Pushed downwards and the target would now be falling faster than the BM.
Now, keep that falling scenario in mind as we translate the same mechanics to the BM and his target on the ground. Going back to the question: would Pushing Attack allow the BM to push someone 15ft in the air?