Hey there SignZ. It wouldn't be hard (theoretically), all that'd have to be done is saving the sound channels into a variable, then saving that variable to a file. The problem here is that it'd be .RAW sound, and to play it it'd have to be a .wav sound. I'm not sure if there's an encoding function available in PALib/ASLib, but even if there isn't, the sound could be encoded on the PC.
For now, if you want to record the sound, I suggest connecting your DS to the Line-In or Microphone port with a mini-jack male to mini-jack male connector and simply recording it as you would with any other sound.
FalloutDS: Postponed until proper libraries are created, 10%
Foxi_Lib: A completely revised version of NitroSprites[FINAL], NitroBackgrounds and NitroText[FINAL], 40%