Yes it is possible with a Florida bankruptcy to keep one property but not the other. When a bankruptcy is filed, you have the option to keep or surrender secured collateral, including real property. So you can decide, with the help of your Jacksonville bankruptcy attorney, for each property that you own what would be in your financial best interest: to surrender the property to the creditor or to retain the property and continue payments. But be mindful that in Florida, you can only take one homestead exemption in bankruptcy.
If you are behind on your payments, this can be cured. You can pay the arrearages that you owe through your Chapter 13 Plan. Talk with a Jacksonville bankruptcy lawyer to decide if keeping both a homestead and rental property would be in your financial best interest.