React-Native Updating List View DataSource

Try this:

pressRow(rowData){

    var newDs = [];
    newDs = this.state.ds.slice();
    newDs[0].Selection = newDs[0] == "AwayTeam" ? "HomeTeam" : "AwayTeam";
    this.setState({
      dataSource: this.state.dataSource.cloneWithRows(newDs)
    })

}

This should make a copy of the array, which can then be modified independently of the original array in this.state.ds.