What would you do if senior management demanded delivery of software in an impossible deadline?

The feasibility which could lead to happen this type of situations is due to critical business needs. It means senior management would be actively involved and here agile framework will come into play. Delivering the workable product and then further developing the solution would be the best possible shot for win win situation.

It is possible to keep scope, resources & timeline (dates) unchanged, but compromise on quality. This will impact team retention, especially the stronger engineers on the team, over the longer run. The leadership team must understand the consequences.