class RefreshableList extends Component {
  constructor(props) {
    super(props);
    this.state = {
      refreshing: false,
    };
  }

  _onRefresh = () => {
    this.setState({refreshing: true});
    fetchData().then(() => {
      this.setState({refreshing: false});
    });
  }

  render() {
    return (
      <FlatList
        refreshControl={
          <RefreshControl
            refreshing={this.state.refreshing}
            onRefresh={this._onRefresh}
          />
        }
        ...
      />
    );
  }
  ...
}