Yes, every government has the power to ban bitcoin in their country. But The only way to ban bitcoin is to ban the Internet in the entire country (north Korea style).
There are already countries who banned bitcoin other than china, so this is just a normal event on the cryptocurrency world. If you guys are afraid that bitcoin will die because China banned it, don't be bother by this. Bitcoin could exist without this Chinese Government, there is some countries already who trusted bitcoin and make it one of their currency, and one of them is Japan, so rest assured, bitcoin will still live and will keeping booming as always and will sooner lead to revolutionary period throughout the world industry.
As of China, government has only closed exchanges. But soon they'll be launching their own cryptocurrency through which people can easily buy Bitcoin. But all the funds coming and going from bank would be under government scrutiny through their cryptocurrency. This is what they wanted to achieve. They never wanted to ban Bitcoin.
Other factors which I think is they can prohibit mining or regulate it so hard that everybody stops mining all together. China did this and banned mining and trading bitcoins recently that's why it is possible that bitcoins can be banned. If the government however does intervene, they will most likely stop mining operations and the bitcoins harvest will not join the circulation, reducing the supply further making it more expensive thus preventing investors to buy more because it's expensive.