Is it possible to enter DFU mode in Arduino NANO v3??
No, the DFU mode refers to a feature of USB capable microcontrollers.
The Nano uses a 328p (not USB capable) with a usb-serial converter chip (originally FT232, in the clones, it's usually a ch340G).
If the ftdi232 were an atmega16u2 it would work.
Is there a Nano-size Arduino with a 16u2? The 16u2 is smaller than the FT232. I wonder if somebody has done that.
No but there's Arduino micro, which has atmega32U4.
Yeah, the Micro is a condensed Leonardo. It is bigger than the Nano.
And then there is Teensy 3.1. Built-in USB and HID, 3V3 I/O, and 5V tolerant. And 256K RAM.
The Adafruit Metro Mini is almost like the Nano, same form factor.
But it too uses the FTDI chip, but very small smd version.
I'm sort of surprised, honestly, that with all the nano clones, nobody's made one with 16u2 for serial.
I'm not surprised. I seldom see the 16u2 used for anything other than USB/Serial. Most users with 16u2-based boards have little idea of its additional capabilities.
In addition to the the Arduino Micro, which is pretty official but slightly bigger than a Nano, and ARM-based boards like the Teensy3 and Teensy LC, there is also the older Teensy2, which has a 32u4 and is even smaller than a Nano (even slightly smaller than a "pro mini")