Smartthings system works when wifi is down?

Smartthings does have limited local processing. You can check which devices and SmartApps are qualified with your system here. Only SmartApps that’s working with local processing is smartthings smart lighting.

Your local devices
https://graph.api.smartthings.com/localDevice/list?offset=10&max=10

Your local smartapps
https://graph.api.smartthings.com/localInstalledSmartApp/list?offset=10&max=10

Edit : here is a community created list of devices qualify for local processing