There's no better example of mobile-first or mobile-only country than Nigeria. According to an independent web analytics firm StatCounter, 88% of the web traffic in Nigeria originates from mobile devices.
StatCounter has a page on their data collection methodology but nothing on individual countries so Nigeria's sample size is a question mark.
Nigeria, known as "the Giant of Africa", is the most populous country in Africa (174 million people), and the seventh most populous country in the world. Its broadband penetration hovers at around 6% despite the $1.6 billion investment in submarine fibre optic cables that offers fast connection with the rest of the world.
June 2013 estimates for active mobile subscriptions stand at 117 million and big mobile manufacturers are targeting the Nigerian market with special, low priced devices.
As always, it's worth treating data sources with a pinch of salt and not read too much into it in terms of exact traffic numbers. The trend however is clear - Nigeria has leapfrogged PC-based internet browsing and is the true leader of mobile-first and mobile-only population.
Nigeria is also a true Nokia country - a whopping 70% of the internet connected phones there come from the Finnish manufacturer (source: Buzzcity)
At a continent level, Africa will change from being mainly PC-based to mobile-based very shortly, perhaps even before 2013 ends. Currently, roughly 43% of web browsing in Africa happens via mobile devices.
This is considerable compared to mobile internet usage in the rest of the world which ranges from 30% in Asia, to 14% in North America, to 9% and 10% in South America and Europe respectively.
mobile internet traffic in South America stands at 9%, in Europe at 10%, in North America at 14%, in Asia at 30%.