final CarouselController carouselController = CarouselController();  <- here

class CarouselDemo extends StatelessWidget {
/// CarouselController carouselController = CarouselController(); <- not here

 @override
  Widget build(BuildContext context) => Column(
    children: <Widget>[
      CarouselSlider(
        items: child,
        carouselController: carouselController,
        options: CarouselOptions(
          autoPlay: false,
          enlargeCenterPage: true,
          viewportFraction: 0.9,
          aspectRatio: 2.0,
          initialPage: 2,
        ),
      ),
      RaisedButton(
        onPressed: () => carouselController.nextPage(
            duration: Duration(milliseconds: 300), curve: Curves.linear),
        child: Text('→'),
      )
    ]
  );
}