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Unfortunately, TGIFImage type is not included into the Delphi. So you must establish a component, which supports this type.
After that you can try this:

procedure TForm1.Button1Click(Sender: TObject);
var
  Bitmap: TBitmap;
  GIFImage: TGIFImage;
begin
  Bitmap:=TBitmap.Create;
  Bitmap.Canvas.TextOut(1, 1, Edit1.Text);
  GIFImage:=TGIFImage.Create;
  GIFImage.Assign(Bitmap);
  if SaveDialog1.Execute then
    GIFImage.SaveToFile(SaveDialog1.FileName);
end;

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