Sony’s official battery for the Sony RX1R III is the Sony NP-FW50, one of which is included in the box with the RX1R III.
A micro USB cable to charge the NP-FW50 in-camera is also included when you purchase a new RX1R III but there is no external charger included.

Here are my recommended battery and external charger options for the Sony RX1R III.
Batteries
My advice would be to stick with the original Sony NP-FW50 battery for the RX1R III, don’t be tempted by the cheaper third-party options that may not work correctly in the RX1R III.
If third-party batteries do work they can often be disabled via future firmware updates.
Original Sony NP-FW50 Battery

The original Sony NP-FW50 battery is certainly more expensive when you compare it to third-party options. However, over the long-term the original Sony batteries do hold their charge better a lot better than the third-party battery options and they are guaranteed to work.
External Battery Chargers
The Sony RX1R III does not include an external battery charger in the box, the only option you have to charge the battery is in-camera via micro USB.
If you would prefer a faster external charger then the Sony BC-TRW W is the correct charger for the Sony NP-FW50 battery.
Sony BC-TRW W Series Battery Charger

The Sony BC-TRW W charger will charge a completely drained NP-FW50 battery in around 220 minutes.
USB Charger
You can also charge the Sony NP-FW50 in-camera via micro USB and a USB charger whilst you are on the move. Here are some recommendations.
Anker PowerCore 20100mAh

Price Check & Buyer Reviews for Anker PowerCore 20,100mAh
At: Amazon
Battery Life
For shooting stills RX1R III battery life will vary considerably depending on how you are shooting and the environment that you are shooting in.
According to Sony the NP-FW50 rechargeable battery pack allows up to 300 images (CIPA standard. Using the LCD monitor. 270 images when using the viewfinder.) to be shot on a single charge.
Sony RX1R III Guides & Resources
Check Price & Reviews: Sony RX1R III
At: B&H Photo
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