Having won an eBay action for a known-dead Sony DSC-F717 at $0.99 (plus $15 shipping, the seller being no fool), I now have a possibly salvageable camera, a Genuine Sony AC supply, and two more NP-FM50 batteries for about the price of any one of the components.
One battery arrived stone-cold dead, suggesting the camera had been put away with the battery installed for a very long time and they died companionably. The camera still charges a (good) battery, even though it doesn’t turn on, and perusing the schematics suggests checking the power switch, because it’s always the switch contacts. That’s for another day, though.
For the record, the battery status:
The red and green traces come from the two batteries I’ve been cycling through the camera since, um, 2003, so they’re getting on in years and correspondingly low in capacity.
The fourth battery (2019 D, the date showing when it arrived, not its manufacturing date) went from “fully charged” to “dead” in about three seconds with a 500 mA load, producing the nearly invisible purple trace dropping straight down along the Y axis.
Sawing the dead battery case around its welded joint at a depth of 0.75 mm, then prying with a small chisel, exposed the contents without histrionics:
Now, there’s a name to conjure with. Turns out Sony sold off its Fukushima battery business a while back, so these must be collectibles. Who knew?
The lower cell is lifeless, the upper cell may still have some capacity. Three pairs of 18500 lithium cells are on their way, in the expectation of rebuilding the weakest packs.
After desoldering the battery tab on the right from the PCB, it occurred to me I needed pictures:
Yeah, that’s a nasty melted spot on the case, due to inept solder-wickage.
Unsoldering the three tabs closest to the case releases the cells + PCB from confinement:
I’m still bemused by battery packs with a microcontroller, even though all lithium packs require serious charge controllers. At least this is an Atmel 8-bitter, rather than 32-bit ARM hotness with, yo, WiFi.
The cells have shaped tabs which will require some gimmicking to reproduce:
Now, if only I could reboot the camera …
The Smell of Molten Projects in the Morning 
Sony’s batteries have a certain sameness in form factor. Your NP-FM50 looks like a cut-down NP-F550, of the type I found in a 1999 vintage Digital Mavica (yes, the one that uses floppies and a ridiculous 15x zoom) scored working from the Michigan State Surplus Store – https://msusurplusstore.com/ – the other week. I was pleased to see that the F550 is still a current camera battery, and cheaper aftermarket suppliers still make ’em.
The photo lamps take NP-F550/750/950 batteries and those NP-FM50 batteries snap right in (with a correspondingly short life). The converse is also true: F550s stick way out of the camera’s hatch and lack the “Infolithium” contact allowing the camera to ignore alien batteries.
I failed to find a table of NP-Fxxx battery characteristics, but it’s obvious Sony just stacked up enough 18550 / 18650 / whatever cells to get the capacity they needed; F970 batteries are absurd, but you’d need ’em for a photo shoot far from an outlet.