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Yes, you can compile a program if you have its source-code, but it's far more difficult to swap the image of a running process on-the-fly. Very often the executable file is locked to prevent such modification. Almost always, it's simply ignored if you try.
When a program is executing, its code-segment is almost always protected against modification, for a variety of reasons not just this one.
(And all of the foregoing comments are "not-at-all Linux-specific.")
Donald Knuth, in the Art of Computer Programming (one of the early parts, if I remember correctly) considers the problem of self-modifying code and is prepared to do it and so has an example. Of course, this is in assembler for the hypothetical MIX machine, but you should be able to extrapolate from there.