Equivalent to POSIX mmap() in AVR-libc
No.

First, mmap() makes almost no sense on an AVR, as there's no memory management unit or concept of process memory to configure.  Some related concepts like sbrk() are occasionally implemented in a minimalist fashion to let library functions designed for larger computers work, but they typically only update a tracking variable or fail.  For things that are physically realizable you can write your own mmap() if you really want, either allowing (assigning) or disallowing a region, and even if you really want automatically copying part of a file off an SD card or whatever...

But more importantly, your actual goal is entirely unachievable, as the AVR is a strict Harvard Architecture which can only execute instructions from flash, never from RAM.

If you want to execute from RAM, try one of the various small ARM Cortex M0/M4/M3 platforms (many of which have Arduino ports), or even consider a larger Embedded Linux system, which will not necessarily be any more expensive than an Arduino.