Abbas, 80, was speaking publicly for the first time since rumours surfaced last week that he was in poor health, which the PA has categorically denied. He did not discuss the matter and appeared well.
He also spoke as three months of violent attacks by frustrated Palestinian youths on Israeli targets have made the PA and its leadership appear increasingly out of touch.
The PA, the governing authority set up under the 1993 Oslo peace accords with Israel , has faced funding shortages, and its ongoing security cooperation with the Jewish state has been heavily criticised.
Israeli Prime Minister Benjamin Netanyahu is preparing a contingency plan in the event of the PA's collapse, according Israeli media reports.
"I have heard a lot of talk in the past few days about the Authority, the destruction of the Authority, the collapse of the Authority," Abbas said. "The Authority is an achievement of ours that we will never give up."
"Don't dream of its collapsing, don't even dream," he told a press conference during a lunch during a lunch to mark Christmas, which some Orthodox churches celebrate on Thursday.
The PA was meant to be a temporary body until a fully independent Palestinian state was created, but more than two decades after Oslo young Palestinians see little hope of the dream becoming reality -- and many do not feel Abbas represents their concerns.
In a recent poll, two-thirds of Palestinians said they believed a new armed uprising, or intifada, would serve "national interests" better than negotiations.
Abbas refused to countenance an end to the PA.
Answering a question about what will happen after the PA, Abbas said "the Authority is here, and after it comes the (Palestinian) state. No one has any other scenarios, and we will not accept another scenario from anyone."