For years, thousands of people in the rolling hills surrounding this city signed leases to permit natural-gas drilling on their property, thinking it was an easy way to earn a few hundred dollars a year.

Now, hundreds are suing to get out of those leases, contending the contracts’ time limits have expired and are undervalued. But gas drilling companies are making a novel argument to keep the property owners locked in: New York’s four-year-old ban on gas drilling using high-volume hydraulic fracturing is akin, legally, to an act of God or natural disaster that automatically extends the leases.

The battle has angered residents who say the companies—including Chesapeake Energy Corp. and Inflection Energy—are exploiting the state’s halt on high-volume hydraulic fracturing to avoid negotiating new leases that pay higher rates like those seen in other states, such as Pennsylvania.