In stand mixer bowl, combine warm water and yeast; let stand 10 minutes. Using a paddle attachment, stir in eggs, banana, oil, sugar, and salt; mix 30 seconds. Add 1-cup flour; beat 2 minutes.
Switch to dough hook attachment. Gradually stir in dates and enough remaining flour to make a soft dough. Knead until smooth and elastic, about 5 to 7 minutes.(NOTE: For hand mixing: following above steps, mix ingredients in large mixing bowl using a wooden spoon or dough whisk. Knead on floured surface until smooth and elastic.) Place dough in lightly oiled bowl and turn to grease top. Cover; let rise until indentation remains after poking dough with finger down to second knuckle, about 1 hour.
Turn dough onto a lightly floured surface; punch down to remove air bubbles. To form a loaf, roll or pat into a 12x9-inch or 12x8-inch rectangle. Starting with shorter side, roll up tightly, pressing dough into roll with each turn. Pinch edges and ends to seal. Place in lightly greased 9x5-inch or 8½x4½- inch loaf pan. Cover; let rise until indentation remains after touching.
Preheat oven to 375°F.
Bake for 30 to 35 minutes. Remove from pan; cool on wire rack.