Thank you for the help!
M3vuv, I did also test using an ESR meter. I use a LC102 and an Atlas ESR+. One of the capacitors was below 1uf, so neither meter registered a reading for ESR. Both the old and the new capacitors tested very well.
James, The only part that resembles the one in your picture is this thing. I believe it's a buzzer/beeper from Murata:
With the boards removed from the machine, I tested various components on each one but could not find any residual DC voltage. Absolutely nothing.
I find no mention of the battery in the service manual.
I was in contact with one of the team members of HP that designed the 35660A, but unfortunately, his area of expertise didn't cover the power/memory portion of the 35660A. He thought the up and down and the right to left arrow buttons were ridiculous! HP went to a rotary control on the 35665A.
I spoke with George at GLK Instruments last year. He specializes in the 35665A. He had no clue where the battery was either.
As for the Floppy drive, this particular one was prone to failure and it was also changed to a different style in the 35665A.
Maybe I just won't shut this thing off