Read the User Manual for device programming.
This is where the subject gets real and hard... Can code run out of RAM? Beats me! It's in the 346 page datasheet somewhere:
http://ww1.microchip.com/downloads/en/DeviceDoc/PIC32MX1XX2XX-28-36-44-PIN-DS60001168K.pdfHow does the device get programmed? ICSP (In Circuit Serial Programming) is one method but there is mention of an In Application Programming method as well. Can either method load code into RAM?
Sometimes code for multiple programs is stored in a large flash device and a second MCU selects which code to run and actually performs the ICSP for the main MCU. Still, we haven't proven that code runs out of RAM and that's a fundamental question.
A related question is whether there is enough RAM for the code and data space. It's not unusual to have an MCU with gobs of flash but not so much RAM. One version of the PIC32MX device has just 256 KB of Flash and only 64KB of RAM (from the datasheet linked above). That doesn't seem like a lot...
Have you got a working game ready to run on the PIC32MX? Will the entire project fit in 64KB? You can have some game independent code in the flash, of course, but even that doesn't seem like a lot of space.
It's one thing to pick a neat chip like the PIC32MX devices, it's another to pick one that will actually work in the application. You definitely want to run code out of RAM, make sure you have that capability before you lock in on a device.