Extreme cold by itself will not cause battery failure. If the battery is discharged for any reason, such as the alternator did not fully charge the battery last time the engine was run, perhaps something was left ON last time the engine was running or possibly even the battery had an internal short, then another condition can enter into the situation. A battery cell can freeze and crack the case if the cell is allowed to completely discharge. Also, if the cell completely discharges then sits for a period of time, the plates in the cell can “sulphate”, making them unable to ever take a charge again. The battery is nothing more than a storage device. It does not MAKE electricity; it only stores what was produced by the alternator. The battery will fail over time, typically a family car that is doing an average mileage and is well cared for should last around five years if of a decent build standard.
At the first sign of failure replace the battery.