The arena has had an ice plant. It is being updated and upgraded
The (Whatever) Field House in Indianapolis doesn’t have an ice plant.
It does. Or at least it used to. The Indiana Ice junior team used to play there. Here's a pic of one of their games in the building.
With that said, the one NBA arena I know had an ice plant but I think may have removed it is New Orleans. The city's old ECHL team the Brass were actually the first tenant of the building, but when the Hornets (now Pelicans) arrived, their lease put all conversion costs for non-basketball events on the other event. That's is they'd have to pay for both going to hockey configuration from basketball configuration before the game and restoring back to basketball mode after, almost doubling their facilities costs. They couldn't afford that, so they left. I think the ice plant was removed after Katrina, as it was no longer needed (notably, when Disney on Ice plays New Orleans, they use the University of NO's Lakefront Arena.)