I have experienced this a few times. The transmission actually stays in first gear in such failure if the mode select switch was left in S. The identifier of this failure is that your gauges will not show the gear position. Its quite odd I know, because the failure is intermittent the TCU does recognize the shifter select once and allows the 1st gear shift to be initialized. Then it goes blind and stops.

This can happen from a very low battery or a faulty gear select lever unit too - In my case the car I was working on was the same as yours and the gear selector was indeed faulty, You need to connect your scan tool and test the EGS module for gear position.

a bit more info is that it showed up on very hot days.

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2012 BMW X5 (Beef + Granite suspension model)

1995 E300D - The original humming machine (consumed by Flood 2017)
2000 E320 - The evolution (consumed by flood 2017)