Preheat the oven to 350 degrees F. Grease an 8-by-4-inch loaf pan with butter and line the bottom and 2 sides with parchment paper.
In a medium bowl, combine the flour, baking powder, baking soda and salt. In a large bowl, cream together the granulated sugar and eggs. Stir in the mashed bananas, oil and vanilla. Stir in the flour mixture a third at a time and mix until just combined. Remove 1 cup batter to a separate bowl and stir in the strawberry preserves and half of the freeze-dried strawberries.
Pour the plain batter into the prepared pan, alternating occasionally with a spoonful of the strawberry batter. Use a skewer to swirl the batters. Bake until a clean skewer inserted in the center comes out mostly clean (some moist crumbs should stick to it), about 1 hour 15 minutes. Let cool completely, then remove the bread from the pan.
Mix together the confectioners' sugar, lemon juice and remaining freeze-dried strawberries in a small bowl. Drizzle the mixture over top of the banana bread before serving.
If you don't have an 8-by-4-inch pan, a 9-by-5-inch pan will work; just cut the baking time by 10 minutes.
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