Combine the bananas, sugar, water and lime juice in a small saucepan over medium-high heat. Stir to combine and bring to a boil, then reduce the heat to low. Simmer for 25-30 minutes, stirring frequently, until the jam is thickened enough that you are able to briefly expose the bottom of the pot while stirring. When this point is reached, remove the jam from the heat and stir in the vanilla. Transfer the jam to a jar and refrigerate.