Whisk together milk, persimmon purée or mash, brown sugar, eggs, raisins, vanilla, and salt in a large bowl, then stir in bread and let mixture stand at room temperature 15 minutes.
Put oven rack in middle position and preheat oven to 375°.
Stir walnuts into bread pudding, then spoon pudding into a buttered shallow 8-inch square (2-quart) glass or ceramic baking dish, spreading evenly. Dot with butter bits. Bake pudding until golden, puffed, and set, 35 to 40 minutes. Cool to warm in pan on a rack, about 20 minutes.
Bread pudding can be made 1 day ahead and cooled completely, uncovered, then chilled, covered. Reheat, uncovered, in a preheated 350° oven until warm.