This one is interesting, and I had to do it a couple of times to make sure it was repeatable.

Normally, if you attack a shipyard or starbase and you take its hit points down to zero, there will be a little explosion animation and the shipyard or starbase disappears.  However, I have found that if I attack a shipyard or starbase using the Antimatter Bomb that is a weapon on my Harpazein command ship (special Mimot Brotherhood ship) and that attack takes the hit points of the target down to zero, there is no animation and the graphic for the target stays visible.  If you hover over it no popup windows appears indicating that the target is no longer there and if you click on the target the Influence window appears indicating that it is an empty cell.

The first screen snapshot below shows a shipyard that I attacked using the Antimatter Bomb.  The shipyard only had 8 hit points left when I attacked it and the Antimatter Bomb does 12 damage so the shipyard should have been destroyed.

The second screen snapshot below shows the Influence window that shows up when I click on the shipyard indicating that the shipyard is really no longer there.

The programmers should have fun figuring this one out!