David,
No it would not, UNLESS the user doesn't click on any more links on the homepage. Really, a bounce is calculated based on whether or not there was more than one pageview of a site in a single session, not whether or not the user left. GA (without doing some custom work) doesn't even know the user has clicked on an external link.
Edit: My apologies. I misread your question. I thought you were saying that the banner click was a click off to another site. Still, the answer stands.
As long as the GA snippet exists on the 2nd page, it will trigger a pageview and look like the 2nd pageview in the session, meaning it's not a bounce.