British publisher Frederic Warburg (who published English editions of Kafka's novels during the heavy bombing of London in World War II) mentions in his book "All Authors are Equal" (he was also George Orwell's publisher) an anecdote involving Brod, Kafka and their manuscripts. He attributes this story ("for whose authenticity I do not vouch") to Hannah Arendt, but does not provide a source. In it, Brod is strolling down a Prague street, a few days after Kafka's death, and meets a literary editor ("Let's call him Rudi," writes Warburg):

Rudi: "You look sad, Max, indeed we are all sad at the shocking news of poor Kafka's death."

Max: "… my friend Franz placed on my shoulders a heavy burden. Franz has given me instructions that I am to burn all his unpublished work, all of it."

Rudi: "Well, you must burn it, then, as Franz wishes."

Max: "It is not so easy, my friend. I have read his work, his novels and stories, all of it. These are masterpieces. How can I burn them?"

Rudi: "Masterpieces, you say. Then you must not burn them, Max. You must have them published."

Max: "Against dear Franz's wishes, Rudi?"

Rudi (thinks hard, then in an emphatic voice): "I have it, Max. Publish Franz's work and burn all your own."

Via a piece in Haaretz. (Via Literary Saloon)