The Tyne Valley structure, known as the wobbly bridge, was previously damaged when trees and debris washed downstream during heavy rain in 2013.
The bridge at Allen Banks reopened in November 2014 after the National Trust raised Â£100,000 to restore it.
The trust said the whole site has been closed to visitors as the entire area is thought to be "unstable".
A 300-metre landslide has also washed away the network of paths at the site, with the full scale of the damage unknown, the trust said.
Ranger Chris Johnson said: "Allen Banks and Staward Gorge is not safe for walkers at the moment. We've seen multiple landslips which have uprooted large trees and washed away footpaths, and the land is still moving.
"We need to wait until the movement stabilises before we can properly survey the site, understand the full extent of the damage and make plans for the future."
The suspension bridge is one of the only surviving features of the wilderness garden created in 1850 by Susan Davidson as part of the Ridley Hall estate.
Allen Banks has been classed as a Site of Special Scientific Interest because of its rich plant and animal life, and the walks along the river gorge through the ancient woodland attract about 50,000 annual visitors.
A restored Victorian bridge that forms part of a woodland trail has been destroyed by flooding.