Is it possible to retrieve the last modified date of a file using Javascript?

If it's on the same server as your calling function you can use XMLHttpRequest-

This example is not asynchronous, but you can make it so if you wish.

function fetchHeader(url, wch) {
    try {
        var req=new XMLHttpRequest();
        req.open("HEAD", url, false);
        req.send(null);
        if(req.status== 200){
            return req.getResponseHeader(wch);
        }
        else return false;
    } catch(er) {
        return er.message;
    }
}

alert(fetchHeader(location.href,'Last-Modified'));

This seems to be useful, and works for me - giving you the 'local' date

document.lastModified 

Compared to the above selection of req.getResponseHeader() it's one less round trip/http call.

Using the modern fetch method:

var lastMod = null;
fetch(xmlPath).then(r => {
    lastMod = r.headers.get('Last-Modified');
    return r.text();
})