Do the power-offs occur if you run the Nano in 5W mode? (sudo nvpmodel -m 1)
If not, then it points to a power supply issue. Even though your power brick is rated for 50W, it may not deliver the required power consistently, or has voltage droop across the cable. I would check out some of the power supplies from this thread or the Jetson Nano Supported Components List, unless you are already using one of these.
Another issue you could check to debug power issue is to disconnect your SSD peripheral and see if the behavior still occurs.