Derek,
As you have a Mark I Beocom 6000 and you only have one phone the most likely cause will be the RF chip, BUT could also be a fault in the base.
The RF chips on the Mark I PCB are no longer manufacturered and so cannot be replaced.
However, B&O do produce an upgrade kit which replaces most of the Mark I parts which you can see HERE
To be 100% sure that this is the fault it would be helpful if you could borrow a working Beocom 6000 to test your base. Having said that, and given your previous statements, I would say the RF chip is the most likely fault.
For you and other members interested in Beocom 6000 issues and variations, there are some common pictures I have previously posted HERE