I think an important consideration is important for whom. If the only reason something is important is because of an arbitrary management deadline, the right response is to stand your ground and calmly finish working on what you already picked up. While I understand it's easier said than done, if you don't stand your ground, it won't get better. The polar opposite of this is when I personally broke something and it's stopping others from doing their work. In that case, the right response is to drop what I was or wanted to be working on and fix the issue.
An extra case of all of the above is planning work around big releases and the like. Regardless of how solid your CI pipeline is, if there's a bigger change set released, it's good to have someone who's designated on call. They're free to work on their stuff if they please, but their main priority for that whole day is the delivery. If anything is fishy, they're the guy who picks it up first.
None of the above happens without good communication. To get to good procedures you need to take the time, discuss with your team, and agree on what your plan of operation is. Make sure it's clearly communicated both inwards and outwards. If anyone internal or external comes to you with a fire, you can point them to your plan. It may be urgent to them, but that doesn't mean it overrides your team's agreements.
With all of the above in place, the scenario you described changes considerably: instead of asking whether you can finish your task, you will instead inform the other side that you're going to finish your task, and if it fits your plan, then deal with their problem. At the end of the day, it's a matter of mutual respect — it doesn't matter what the org hierarchy is, they're asking you for help, they can't demand it.