I find being transparent with my team makes it a lot easier. There are internal deadlines and external deadlines. Former are internal ones which have no real consequence if missed and just keep a development rhythm (e.g. weekly iteration, monthly release, quarterly roadmap update catchup etc...).
The other type comes from third parties, generally, such as clients or the board. Once you separate these two types of deadlines, I find my engineers more understanding about having to meet external ones whereas blurring internal and external into one "deadline" simply causes people to either burnout or lose faith in management because everything is marked as urgent.
I've found most deadlines I've worked with to be internal. The only exceptions to that however was legislation dates; Cookie walls and GDPR, new laws for stock trading on an European level (forgot the name), things like that. These are things for which your company can be fined heavily for. Other deadlines are likely to be of the "we'll start earning money later" variety.
Those are BS as well. No one is going to fine company on day 1 of a new law. GDPR was effective 25 may 2018 but companies still had half of a year to implement it.
Basically if you prove you are working on it I bet they will let you off the hook.
I work on a lot of features lately handed down from the CEO of our client. And he places deadlines so unrealistic his own team oftentimes doesn't get us the details until a couple of days before the deadline. For work that takes weeks to do.
Stuff is constantly pushed back. The India launch so far has been pushed back 9 months. A new product launch in the USA was pushed back 3 weeks - he initially said they were launching January 1st, when they get 2 weeks off for Christmas and no one works on new years day. Lol.
It sounds like management at your company isn't doing a good job pushing back on these requests or insulating the dev team from this unreasonable client?
But you work for your CEO, not your client's CEO. Even Steve Jobs called Larry Page, not a random engineer, when he didn't like the Google icon on the iPhone.
The other type comes from third parties, generally, such as clients or the board. Once you separate these two types of deadlines, I find my engineers more understanding about having to meet external ones whereas blurring internal and external into one "deadline" simply causes people to either burnout or lose faith in management because everything is marked as urgent.