Nano onboard "test" LED blinks when powered via 5V pin
So, as per the comments on the question here's what I think happened:

That "test" LED lights briefly when the board is first powered on, indicating I suppose that it's booted and setup() is either complete or about to run.

It also lights briefly when the reset button is released.

In the sketch I'm using one of the digital pins is connected to RST -- the arduino is being controlled via UART.  I have swapped some of the pin numbers and wiring around, so I tried leaving the one driving RST low instead of high and indeed, this seems to be the issue: It's resetting at ~1s intervals.

(I honestly believe I checked this before by removing all connections, but since I did not note that in the question, perhaps I never did...)