Caching/memoizing promise results
Code looks fine - but it strikes me as somewhat unnecessary.

A promise can only be resolved/rejected once, so if you just keep the promise object itself around, its result is automatically cached within. I.e. if you call .then() on an already-resolved/already-rejected promise object, it'll just immediately invoke your handler(s) with whatever data it was originally resolved/rejected. It won't re-do whatever caused it to be resolved/rejected. So just keep the promise object itself around and you can use it whenever you want.

In order words, caching promise objects makes sense (e.g. caching web request promises), but caching promise results is built-in and happens automatically.

So to that end, it'd be smarter to add caching at a higher level. I.e. don't wrap generic promises, instead wrap specific tasks like network requests and such where caching the promise is desired and you can have more control over what keys to use, when to invalidate the cache etc..

I'd do something like this

// *specialized* function for, say, network requests,
// where caching is desired. In this case, we're caching
// promises by the URL they're fetching
var fetch = (function () {
  var cache = {};
  return function (url) {
    cache[url] = cache.hasOwnProperty(url) ? cache[url] : new Promise(/* request magic goes here */);
    return cache[url];
  };
});


(Obviously, you'll want a way to invalidate the cache etc., this is just to illustrate the point)