anyone knows miniature wireless system with data rate >= 2000 Hz?
For serial RF over a short range (&lt; 5 meters mentioned by OP), you can find TI CC2500 based standalone low-cost, low power modules with an ADC built in, such as this one. A search on eBay and similar sites would find you other such modules based on this IC. The size is well below the specified dimension constraints.

The device essentially takes TTL and analog inputs and pushes them in serial data form through the wireless channel to configured receivers (again, the same module can be used) at speeds up to 38.4 kilobaud. Power consumption is extremely low, but so is the sophistication of the device: It is basic but gets the job done.

An alternative IC to search for, on which similar ADC + serial RF modules are based, is the nRF9E5 which has a built in 8051 microcontroller and ADC. 

Another, popular option is an XBee or ZigBee module. These come in a variety of configurations, from basic serial packet radio through Bluetooth and WiFi. 
Here is one such option.

Search for "Xbee Direct IO" to find suitable modules. This YouTube video shows one such module in use.

All these options are smaller than the 3 square inch size constraint, and most do not require external antennas, they use a PCB trace as an antenna.

Do check the ADC specifications in each case for your specific purposes, though, as there is a wide range of options on that front.