I'm aware that 行かない means "don't go" (hope I'm not wrong).
What I don't understand is the function of で. What's its use in this sentence? (I usually see that character used in です).
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いかないで, without any other context, is the short way of saying いかないでください, "please don't go". A Japanese verb ending with the positive て or negative ないで as a single, one-word sentence, is a "please do / please don't" situation:
走らないで（ください）(please) don't run
座らないで（ください）(please) don't sit down
喋らないで（ください）(please) don't talk
omitting ください is often done in casual conversation. It can also give the request a strong feeling when it is omitted.