Sitemap for a one-page site

I'm assuming you're referring to an XML Sitemap? If so, then no - the sole purpose of an XML sitemap is to facilitate in the discovery (for indexation) of URLs/Pages. So if you only have one page, then there's no point. If you're worried about being found by search engines, you can either link to your page (from another site, or a twitter/Google+ page), register your domain with webmaster tools (either Bing or Google).

On the off-chance you were referring to an HTML sitemap, then no - your single-page site becomes a two page site, and you only have internal fragment (hash) links on your main-page anyway.

There's an exception here, and that's if you have videos on your site.

You can create a separate sitemap just for video, or you can embed a video sitemap within an existing sitemap, whichever is more convenient for you.

The same can be said for for images and news if applicable.