For an integer an independent k-rainbow dominating function (IkRDF for short) on a graph G is a function g that assigns to each vertex a set of colors chosen from the subsets of satisfying the following conditions: (i) if , then , and (ii) the set is an independent set. The weight of an IkRDF g is the value . The independent k-rainbow domination number is the minimum weight of an IkRDF on G. In this paper, we initiate a study of the independent k-rainbow bondage number of a graph G having at least one component of order at least three, defined as the smallest size of set of edges for which . We begin by showing that the decision problem associated with the independent k-rainbow bondage problem is NP-hard for general graphs for . Then various upper bounds on are established as well as exact values on it for some special graphs. In particular, for trees T of order at least three, it is shown that .