We are dynamically serving a HTML page that is appropriate for mobile users. It has the same URL but different HTML source code. We also have a separate mobile site.
Is there any problem for Google SEO with two different sites that both do mobile?
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Google have a guide here as to how they handle adaptive and responsive design, including "Dynamically serving different HTML on the same URL":
In your case, it sounds like you just need to add
Vary: User-Agent to your header.
Yes there might be problems due to Google penalizing your site because you are showing different content for different clients on the same URL. While the chance is not high, there is still a possibility of this. If you want to avoid it, I would suggest redirecting mobile users to a mobile site instead.