Battery solution Nano 33 iOT
You can either use the 3.3V pin to supply power to the board, or use the Vin pin.

By using the VIN pin to power goes through a (step down) voltage regulator, that provides a nice stable 3.3V to the board. The regulator in the Nano 33 IoT is the MPM3610, which requires an input voltage of at least 4 - 4.5 Volt. Which is more that a LiPo or LiFePo single cell can supply. You could use a multi-cell battery pack. Or a USB power bank. So this solutions is not really ideal.

The other option was to connect the battery directly to the 3.3V pin. The voltage of a LiFePo however changes as it is being discharged. The voltage is around 3.65V when full, and 2.5-2.75V when "empty". The 3.2V rating is called the nominal voltage (kind of an average voltage).
So it's a bit too high when full, and too low when empty. The voltage also isn't regulated, so it can fluctuate a bit, based on the amount of current that's being used.

The maximum voltage for the SAMD21 on this Nano is 3.63V. Maximum voltage for the gyro (LSM6DS3) is 3.6V. The absolute maximum voltage for the radio (NINA-W10) is 3.6V.

So a full LiFePo is slightly too high for the Nano. Though I think you should be fine. But don't try this with a regular LiPo of Lion battery (4.2V when full).