Wifi on Nano v3.0 ATmega328P-20AU Board
The described requirement seems to fit the BeagleBone Black:

 

The board is small (86.36 x 54.61 mm) though not as small as the Arduino Nano. 

It has 2 USB ports, one of which is USB Host Mode, which will be essential for the WiFi dongle. The second USB port can be used to power the board. 

The BBB natively runs Angstrom Linux, and also supports BBB-specific Ubuntu and Android builds. Users can SSH into linux running on the BBB. 

Several tutorials on using Python on the BBB are available, such as this one for IO Python.

In addition, WiFi "capes" (add-on boards) such as the WL18xx are available for the BBB. This may be a better-integrated option than using a WiFi dongle, courtesy the native support built into the BBB Angstrom Linux distribution. 

The BBB offers the best of both worlds: High level use via Linux running on the main processor, as well as extensive low level functionality including GPIOs and timers based on the two 32-bit low-level PRU microcontrollers on the board. This provides several advantages over the Raspberry Pi, which somewhat restricts low-level hardware access.

The BBB board does have an HDMI port and a graphics accelerator on board, as well as a floating point accelerator, but those may be ignored if not needed.