Cannot access array of JSON objects as props

I have a React app that consumes a nested JSON object retrieved by an sync AJAX call. On return of the call, I traverse the JSON and bind each of the packets of data the components need to individual state variables.

I then call the components in render - also nested - and pass them the data they and their children need as props.

Passing individual values works well via <SubComponent dataToUse={this.props.value} /> However when I pass an array and attempt to .map that array in the component, the browser console gives me the error:

Cannot read property 'map' of undefined

...which obviously seems that the array is not actually there - since it is in render, perhaps it is running map before the AJAX call returns the JSON file? However, if I stringify the array data when binding it to state and pass that, the whole array of data is there, as a string.

If I take that string and parse it back into JSON, I get this:

Unexpected token u in JSON at position 0

Which again suggests that the data is NOT there.

Suffice it say, I am confused. I can easily pass single data variables down my app hierarchy as props but cannot access arrays.

Has anyone else had this sort of problem?

Edit; some code:

My top most component makes an AJAX call with JQuery:

$.ajax({
    type: 'GET',
    url: "/api/" + eventCode,
    dataType: 'json',
    cache: false,
    success: function(data) {
        this.setState({updateList: data.updates});
        }.bind(this)
    });

This AJAX call sits in its own function and is called every 30 seconds here:

componentDidMount: function() {
    this.loadCommentsFromServer();
    var refreshInterval = 32000; // take this from the blob!
    setInterval(this.loadCommentsFromServer, refreshInterval);  
},

The render function is as follows:

render: function() {
    return (
        <div id='App'>
            <UpdateList updateData={this.state.updateList}/>
        </div>
    )
}

The UpdateList component is as follows:

var UpdateList = React.createClass({

    render: function() {


        return (
            <div id='UpdateList'>
                Update list:
                <ul>
                {
                    this.props.updateData.map(function(update, i){
                        return <Updates singleUpdate={this.props.updateData[i]}/>
                    })
                }
                </ul>
            </div>
        )
    }
});

If add a simple test array in getInitialState, the array is mapped and the Updates component renders each item just as desired.