Half of my work are orders from my direct manager, the other half requires that I address specific needs across the whole company - mostly lateral in authority but also a few senior managers. My direct manager is not entirely responsible for any of the other areas, so my performance reviews are similar to 360 degree surveys.
I can postpone or decline other peoples' requests, but in order for me to be successful, build great relationships and contribute positively, I am supposed to satisfy everyone where my judgment says it is beneficial for the company.
Unfortunately, there are situations where work from my direct manager takes all of my time and I am overwhelmed by any external requests. I am physically unable to take on more tasks (sleep, stress, hunger, some bad symptoms).
My manager says I should simply ignore or postpone any other requests, but I am concerned this might ruin other people's perception about me and obviously my professional relationship with them.
What is the best way to communicate laterally/externally that I am overwhelmed, without damaging those relationships?