* Unemployment rises to 11.1 pct in May
* Debt crisis, austerity likely to drive rate higher
BRUSSELS, July 2 (Reuters) - Joblessness in the euro zone rose to a new euro-era record in May, pushed higher by lay-offs in Austria, France and Spain as the 2-1/2 year debt crisis continued to eat away at the fragile economy.
Around 17.56 million people were out of work in the 17-nation euro zone in May, or 11.1 percent of the working population, a new high since records began in 1995, the EU's statistics office Eurostat said on Monday.
Despite an agreement by euro zone leaders at a summit last week to help stricken banks and try to bring down unaffordable borrowing costs for southern Europe, the euro zone's economy remains stuck in a deep malaise.
Companies in the euro zone have frozen or cut back spending as concerns about the future of the currency bloc saps confidence and in turn pushes them to keep labour costs down.
Weak demand from European households, as well as from the United States and Asia, is also taking a toll. A German-led drive to cut deficits is forcing governments to shrink spending.
Unemployment rose by a tenth of a percentage point in France, the euro zone's second largest economy, to 10.1 percent, while joblessness in Spain, the worst in the bloc, rose again to 24.6 percent from 24.3 percent in April.
Even in wealthier Austria, the number of people out of work ticked up by two tenths of a percentage point to 4.1 percent in May, although that puts unemployment back at January levels.