Possible to only re-render one prop of component in react-native, not whole component? I know that it is possible to re-render only one component (instead of whole DOM) using shouldComponentUpdate(), but is it possible to only re-render one PROP of one component? In one instance in my app I do not want the entire component to update, but only a specific prop that is being changed.

You probably can't do that because that's not how it's supposed to work. The component is updated in order to reflect all changes made to its state and props. The best way to get around this would be to just not update the props being passed by the parent component. 
Alternatively, you could also use the state to hold the values you don't want to be updated, and set these values in the constructor of the component.