Is there a minimum number of years that we must retain proof of BSA-related training & materials?
I know CTR and SAR is 5 years, but not sure of training records.

There is nothing in Appendix P of the FFIEC Exam Manual on record retention for training. That said, 5 years is probably a safe timeframe. At LEAST, you would (obviously) want to keep the information back to your last audit/exam as training will generally be included in testing procedures.