I'm looking for a radio that has the ability to be encrypted using an analog signal.
You posted this in the amateur radio forum. Using encryption/scrambling is not permitted by the FCC with the exception of "telecommand" of amateur radio satellites.
My goal is to be able to talk in mountain areas without any trees or terrain interference.
You need to give us some more information on how much range you are expecting. Topographical features are going to impact radio coverage unless you go down into the HF bands and bounce stuff off the ionosphere. If you are looking for a VHF or UHF radio to do this, you'll have some challenges.
I have tried digital encryption in these areas and it doesn't work very well.
Encryption doesn't impact range of the signal.
Does anyone know of a handheld and base radio that will encrypt analog?
You won't find amateur radios that will do encryption or analog legally.
GMRS, MURS, FRS don't allow it either.
If you have a business type FCC license, you can run encryption/scrambling all you want.
As for which kind of radio, you need to give us more information:
What kind of FCC issued license do you currently hold, or are you willing to get?
What kind of range do you need? Be realistic about your needs, more range = more money.
What is your budget?