[gmiller123456]

There are a lot of people who've shown how to use the ESP8266 powered off batteries for a long period of time. This [1] references a few and contains another itself. But they're not running all the time, it involves going in to deep sleep, and waking up very seldom.

If you don't need actual WiFi, and instead just need wireless communication, there are other lower power modules you can use like the nrf24l01, LoRa, and other generic "Wireless Tranceivers". If you need them to have internet access, you can create a gateway which the remote modules communicate with, then have a ESP8266 plugged into the wall to proxy the requests.

[1] https://www.youtube.com/watch?v=I3lJWcRSlUA