Small Arduino with Bluetooth HID interface I am trying to create a bluetooth gamepad for my phone. 

I saw this:

https://www.amazon.com/DFRobot-Beetle-BLE-Arduino-Bluetooth/dp/B014KQHEH6/ref=cm_cr_arp_d_product_top?ie=UTF8

It says its bluetooth HID, but there are some conflicting comments saying it is serial over bluetooth. Is there something similar in a small form factor with an hid bluetooth module that I can use?
Based on the fact that the controller you mentioned has a TI CC2540 Bluetooth processor, I have found a few webpages with information about HID capabilities in the CC2540. Based this the CC2540 should support HID.

Some example code.
Question on the TI forums, with a link to example code.