Assuming a website offers a great service on one unique page, will it be negatively impacted regarding ranking because this website URL only has one page? In other words, is there a ranking benefit in having many pages in a given website? Or does it not matter, ranking is always per page, not per website?
Ranking is per page. Indeed, when you search something on Google, it shows different pages (different URLs): one page = one URL.
However, you can rank well with service with only one page but in that case, you can't optimize SEO with internal links...
Big websites are good chances to rank better because keywords are displayed on many pages that are linked each other. Moreover, Google like crawling often updated websites. With only one page, it's hard to say to Google you often update your website (or webpage).
In your case, you can read this to create a perfect page for SEO.
A website that has more pages on a single subject generally is much easier to rank since Google will treat that site with more authority. So yes the more content you have the better but this doesn't mean you can't rank a 1 page site.
How To Rank a 1 Page Site
Authority is Key
Ensure that the one page site your making contains information on as few subjects as possible. The more you try to cover more on one page will only dilute the authority. So for example if your site is about Cats, try not to speak about Dogs too much and never talk about your Ford Car for etc.
Ensure that you have unique text and lots of images on the page, since your site is only one page you really want a lot of information on that page, however don't bloat it just for the sake of the rankings. SOMETIMES! LESS IS MORE! You may need to check your site more regularly for content copiers.
Making Your Content Super Easy to Read
With making big sites its often a daunting task to make sure your pages are easy to read, lucky enough you want to create a one page site so this is rather easy. Ensure that you check your pages for Readability before going live as this can help rankings.
Don't Forget the Key Elements
You should ensure your page contains the key elements such as:
HTML5 Can help!!
While HTML5 does not majorly help SEO at present it does help and may help more in the future. Since your site won't have many pages you can easily use good HTML5 and Rich Snippets. A typical example of how you should proceed with HTML5 would look something like...
You can opt to use a menu at the top which uses a smooth scroll to elements of the page, this is considered good for internal anchors which may help your SEO.
Google Loves Fresh Content
Normally Google loves sites that regularly build content, since you are using one page you should try and revisit your pages and spice them up regularly this can be from one week or up to a month depending how much time you want to spend on it. It's worth mentioning that this isn't essential but it may help.
Social Media Signals
Social media can be a great way of getting direct traffic and back links to your site, it may also help your Trust-ability with Google.
Backlinks - Less Requires More
It is possible to rank sites with THIN content but it just takes more, so it is possible but you will need more quality back links. Since your site is small you should focus primary on providing a rich experience with content that people want to link too. If you can't get this right then your site will fail.