@Pseudomizer , Is the esp8266 on a breadboard or is it part of a commercial product like HUZZAH or nodemcu ? I ask because the esp devices can reset themselves from a brownout condition during the WiFi transmit if the power supply circuit is not up to handling the momentary 'high current draw' requirements of the radio. From reading www.esp8266.com, breadboard circuits have caused many hours of troubleshooting enjoyment with the esp8266 parts.
I replace the battery packs on these with 2 Amp cell phone charger power supplies. I have 8 under power 24/7 for months with no problems. They are available from multiple vendors.
http://www.aliexpress.com/item/Free-shipping-ESP8266-ESP-12-serial-WIFI-Industrial-stable-version-A-full-test-board-Full-IO/32260095422.html?spm=2114.01020208.3.49.0YGvDs&ws_ab_test=201556_7,201527_4_3_2_1_4_71_72_73_74_75,0_0