@Larry, Brian, DMR
In the meantime I have replaced the IP5303. Also the 10?F cap. But there is no improvement. The Nano still won't start on its battery.
The battery can be charged normally. I have placed the switch on pin 5 of the IP5303, so that I can start with the key-pin of the IP5303.
All proposals here do not improve. I also replaced the blue LED with a red one. A slightly increased load (+ 50 mA) with an additional LED or a resistor does not cause the IP5303 to start. All measured values ??appear normal.
But after 2 days there is a new phenomenon: after starting the Nano (via the IP5303 pin), it drops out after 7-8 seconds! I can repeat this endlessly, but it is always the same. Also with a full battery, also connected with the USB cable!
This way the Nano is no longer usable. I have no idea where this is coming from. There is no timer in the IP5303 that causes this? I have compared different schemes from IP53xx to IP5306, but they are all the same. The only difference I can find is that the coil (2.2?H) sometimes has a different value.
What could partly help me is that someone would post a detail photo of the components around the IP5303. Other experiences are also welcome.