I searched the web and Chrome Web Store unfortunately there are no extensions to do that.
Only when I tried to make a script I discovered that the things are not that straightforward because there are a lot of nested shadow DOM elements, nevertheless I managed to write it.
Below is the code you need to paste in console(Ctrl+Shift+J) to get all the links.
ditems = document.querySelector("downloads-manager").shadowRoot.querySelector("iron-list").querySelectorAll("downloads-item");
var div = document.createElement('div');
[].forEach.call(ditems, function (el) {
var br = document.createElement('br');
var hr = document.createElement('hr');
div.appendChild(el.shadowRoot.querySelector("#url"));
div.appendChild(br);
div.appendChild(hr);
});
document.body.innerHTML=""
document.body.appendChild(div);
document.head.style.innerHTML="";
After this you can save the resulting page with Ctrl+S.