Can local nations leave the council due to panic?

Nations leaving is a percentage chance. There is a maximum number of countries that can leave each month (based on difficulty) as well as a percentage chance to leave at 5 panic. Having satellites in the country (and neighboring countries) reduces the chance of the country leaving the XCOM project.

To answer the title question: Yes, it is possible for local nations to leave due to panic. However you have inherent bonuses from nearby satellites that reduce the chances of this happening.

Source: INI file and analysis mentioned here: http://www.reddit.com/r/Xcom/comments/1aadl8/probabilities_of_nation_leaving_the_council_at_5/

It absolutely is possible for nations in your continent to secede.

My guess is that this was caused because 3 other nations left at the same time, since only 3 can leave per month on classic difficulty.