In a large mixing bowl, combine the mashed bananas, granulated sugar, brown sugar, lemon juice, dark rum, vanilla extract, ground cinnamon, and salt. Stir until well combined.
Transfer the mixture to a large saucepan and cook over medium heat. Bring to a gentle boil, stirring frequently to prevent sticking.
Once boiling, reduce the heat to low and let it simmer for about 30-40 minutes, stirring occasionally, until the jam thickens and reaches a spreadable consistency.
Remove the saucepan from heat and let the jam cool for about 10 minutes.
Pour the jam into sterilized jars while still warm, seal tightly, and allow to cool completely at room temperature.
Store the jam in the refrigerator for up to 2 weeks or process in a water bath for longer shelf life.