I agree with @jdebari, users won’t likely ask for mobile specifically although they may expect it and think it’s broken. Users don’t and shouldn’t be asking for features, users have problems, not solutions.
That being said, I get why you’re asking the question. If the PM is basing users “asking” for it based on analytics of existing applications, then it’s a fair priority. Working in a specific industry and maybe within a specific context can dictate the value of designing for mobile first. Realistically, are users even allowed to use mobile? Can they access the web app from personal devices? Do you have analytics telling you usage?
I would also break the question into three separate questions:
- As a product team, should we create/use a responsive solution?
- For the design, should we design for mobile first as a priority?
For the first one, the answer is always yes. For the little bit of extra effort, creating or using a pre-built responsive templating system like Bootstrap will save you heartache in the future. You’ll get benefits even if every user is sitting on a desktop computer.
For the second one, this really depends on your usage statistics, user research, and application type. If users aren’t allowed to or the usage is only on large screens, and the application type doesn’t make sense for mobile, it could be detrimental to design mobile first. Certain mobile design conventions just aren’t optimal for a highly interactive desktop business app, a wall display, systems control dashboard or other myriad of uses for a web application. I highlight user research above because you really need a solid understanding of your users and question if your PM’s making assumptions with how users access the application or has evidence to support their decision.