I'm pretty well aware of the discussion between the pros and cons of trying to make a website act too much like a mobile app. Issues with fixed positioning, browser responsiveness, and general network latency inherent in a mobile connection all play against most attempts at this approach.
However, in a recent project at my day job, I think I'm getting pretty close to a slick and seamless mobile web app experience that closely emulates a native one. See the work in progress at: https://staging.ctmc.org. Keep in mind, this is not the final product yet. I will update this post with a new link when it's fully baked.
One of the main downsides to this project in terms of speed is that the site is fully responsive. While this is generally seen as a feature (and I would argue that it is in many respects) one major issue is that more resources are loaded to the mobile device than necessary. Even though it uses a "mobile first" approach in terms of CSS, there are other resources that are present on the page (larger images for the banners, some scripts that don't get utilized on mobile, etc) that play into a longer than necessary page load time.