Screen shots, please, with the layers panel open, expanded and visible, and the object selected.
It must be one or more of these:
On a master page; (I know you say you "unlocked the master" but I really don't know what you mean by that.)
On a layer that is locked; (see Layers panel)
Locked at the object level; (choose Object > Unlock All on Spread)
Thanks John! I'm a novice with Adobe InDesign, so it would've taken me a while to find the tutorial on "Unlocking All on Spread." That did the trick!
Happy to help. I'm glad to know you're deleting borders. You may be a novice, but in most cases, borders are the stuff of amateur design.