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My name is ***** ***** I am a professional here at Just Answer. I noticed that your question had not received a response and thought I would see if you still need help with this.I apologize for the delay and I hope I can still assist you with this here.
If you have not had the wiring checked leading from the TCM to the transmission this needs done.If any issues are found with the wiring this will need resolved to correct this fault.If this is or was checked already and found o.k. let me know so we can continue.
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4F85 code is a faulty transmission control module (located inside the transmission) according to BMW documentation. There were some issues where this code was a result of a bushing failure on the input shaft in the transmission but unfortunately I dont have the document info on this here. The entire shaft needs usually needed to be replaced which is arond $500 to have done.
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It is the TCM inside the transmission which I believe is located on the valve body.
I dont have the tolerances or inspection procedure on this shaft myself or bushing unfortunately.You may have to have this inspected by a shop to see if the shaft needs replaced.
Yes for sure,this will cause excessive play which in turn can cause the transmission to operate incorrectly and cause this issue here.If the TCM is fine and wiring to the transmission is fine as well then this usually indicates an internal mechanical fault needing repaired and the bushing/shaft is most common with this code set here.