I'm in Zone 7, in Arizona. Around here, Vinca minor is such a successful ground cover that's it's considered invasive.
It stays low, is green all year, and has nice periwinkle-colored flowers from time to time.
It does tend to get out of hand, I cut it to the ground with the string trimmer at least once a year. It always comes back strong. I also dig out the strays to keep it from taking over. No issue at all with snow and cold, or drought. It might die back a little, but nothing seems to kill it.
Main downside I see is mice use it for cover. Keeping it cut back does help with this.