MacOS – Cannot Add, Remove, Move, or Drag icons from Dock

I'm using OS X Yosemite (10.10.2) and recently I have been unable to drag or move any items on the dock. Right clicking them gives no option for "remove from dock", and the icons cannot be rearranged. New icons cannot be added, old icons cannot be deleted.

I have tried restarting, and resetting dock preferences to default, and killall Dock, as suggested by earlier askers. None of it works.

I heard about problems with PushBullet – but I do not have PushBullet. Just in case, I deleted a lot of random applications. Still nothing works.

I've tried various commands from lots of answerers on other questions. I've consulted the Mac Help pages. Nothing.

Does anyone know how I can resolve this? After resetting my Dock in an attempt to fix it, it is now filled with the default icons too. This is really limiting my productivity.