Apple - Links in Gmail won't open in Chrome

iOS 10 isn't the problem - the "old" gmail app still opened links in chrome well after updating to iOS 10. This didn't become an issue until updating to Gmail's version 5 that this became an issue.

I'm not sure if the following procedure is exactly what you tried, or if you did something differently – but maybe this helps anyways:

• Run the Gmail app.
• Click the little menu icon in the upper left (three parallel stripes).
• Click the Settings gear icon.
• Click "Google Apps" near the bottom (you may have to scroll down).
• Turn "Chrome" on or off at your preference. On means links will go to Chrome; off means they'll go to Safari.

…as described here.

Update: sorry to say, but it looks like this is actually no longer possible in iOS 10: "iOS 10 won't let you select default apps for third-party apps that handle email, calendar, web browsing, …." (from here)