Partly because our team is overwhelmed, my manager asked me a few months ago to turn down any requests for help or information from the rest of the company unless extremely critical to our own job.
The company heavily relies on our team, though there is a mix of projects taking multiple days on a longer deadline and short requests that could take 1-2 hours to do but are required at a shorter notice. My boss still adamantly says we should use all our time on the main projects rather than the rest of noise. He also asked me to literally reply to all such e-mails with "Sorry, but no".
My concern is that this will start putting me in a negative light across the company. My immediate job is obviously with my direct boss, and I agree that is the most important, but for the longer run, to get favors, to be promoted, be noticed, gain visibility etc. isn't it important to show I am cooperative, a teamworker and helpful across the whole company?
For the past few months, all I have been telling to other colleagues is "No, sorry" or "Sorry, no".
So, is there a way to always say "No" without sounding like I cannot do anything or am always turning down / am unprofessional / unwilling to help / useless to the broader company?
I myself am a manager with reports and I feel that if I want to grow within the company, I need to have some political power - but how to have any power if all I say is "no"?