On the Pico and WiFi boards, the actual VBat pin in the STM32F4 is connected straight to VDD - I'm afraid there is no easy way to power just that and not VDD as well.
There are schematics available (linked from the Pico/WiFi pages) so you could potentially cut the track and wire up VBAT separately - but it would be very difficult
However you might find that entering the proper standby mode of STM32 gives you the power requirements you need. It's probably around 2 uA vs 1 uA for vbat mode - by comparison the MCP1703T voltage regulator on board draws 2 uA, so you're not going to see a massive difference in overall consumption between the two modes.
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On the Pico and WiFi boards, the actual VBat pin in the STM32F4 is connected straight to VDD - I'm afraid there is no easy way to power just that and not VDD as well.
There are schematics available (linked from the Pico/WiFi pages) so you could potentially cut the track and wire up VBAT separately - but it would be very difficult
However you might find that entering the proper standby mode of STM32 gives you the power requirements you need. It's probably around 2 uA vs 1 uA for vbat mode - by comparison the MCP1703T voltage regulator on board draws 2 uA, so you're not going to see a massive difference in overall consumption between the two modes.