If I have a settlement with many roads connected, let’s say at least 6 roads deep, and my opponent cuts it off and builds a settlement between them, can I still build a settlement at the deep end even though my roads aren’t connected to my settlement anymore?
From the rules:
- You may only build a settlement at an intersection if all 3 of the adjacent intersections are vacant.
- Each of your settles must connect to at least one of your roads.
If you don't break either of those rules and you have the resources you may build a settlement.
You might be getting confused with how a player 'breaking' a road with their settlement can shorten the longest road. Another player building a settlement on your road doesn't prevent you building a settlement if it is not adjacent to where you want to build.