That would be the United Arab Emirates.
They are basically conservative because of their religion, but they are as close to 100% non-Muslim as a Muslim country can be.
The city of Dubai is only 10% citizen and 90% immigrant workers as far as population.
I visited the city perhaps 20 times, and when I was there I heard many of the calls to prayer, and I never saw a single person respond. Not one.
Women from all countries are allowed to wear their native dress, including Americans who dress far more immodestly than they could ever get away with in Saudi Arabia.
These people thumb their noses at both Shiites and Sunnis. They're in it for the oil. Their economy is driven by oil production, refining and export. They are basically the Saudi economic right arm.
Also, the value of Emirate currency, dirhams, is artificially pegged to the US dollar.
They have brass ones. The tiny nations flagrantly supported the US for the entire length of the embargo against Iraq (1991 to 2003) and our invasion of Iraq in 2003. Sometimes this meant allowing the US to detain ships from all other Muslim countries trying to run our blockade. (I know this because my ship towed one of these ships into Jebel Ali, an Emirate port).
For Muslims, these are all liberal stands. To us, they're more conservative. (Abu Dhabi, Dubai and Jebel Ali are the most commonly known cities in the Emirates. Jebel Ali is know for having the world's largest man-made jetti system. You can see it built out into the gulf from google earth.)