Can Rum-balls be made without alcohol?
Can you make rum balls without rum?  Well... I suppose you could use rum extract, which has a very intense rum flavor.  You would also want to add some sugar syrup or water to make up for the lost moisture in your recipe.  However, you would not get the exact same outcome due to the lack of the evaporative effect of the alcohol when eating the confection, and due to the lack of alcohol as a flavor carrier.
I surveyed a number of rum ball recipes.  For yields of about two dozen to five dozen, none of them used more than 1/2 cup of rum.   This means that the rum per serving is on the order of a tablespoon teaspoon or so, or less.  You and your friends might feel this is a level of rum consumption that is fine.  I don't drink alcohol myself, and this level would not bother me.
There are many, many, many desserts and confections that you could make.   If your friends are strongly opposed to any alcohol at all, why choose this one, where you are trying to substitute for the key ingredient that gives the dish its identity?