Let r≥0 be a fixed integer. If F is an edge set of a connected graph G satisfying the minimum degree of G−F being at least r, then F is a conditional faulty edge set of order r. The graph G is called F-strongly Menger-edge-connected if each pair of vertices u and v are connected by min{degG−F(u),degG−F(v)} edge-disjoint paths in G−F, where degG−F(u) and degG−F(v) are the degrees of u and v in G−F, respectively. A graph G is m-strongly Menger-edge-connected of order r if G is F-strongly Menger-edge-connected of order r for every F⊂E(G) with |F|≤m and F is a conditional edge fault set of order r, and the maximum value of m is written as smλr(G). Hamming graph KLn has been widely concerned by researchers due to its excellent properties such as good connectivity, scalability, symmetry and iterativity. This paper considers various sufficient conditions of KLn to be F-strongly Menger-edge-connected of order r and determine the exact value of smλ(L−1)r(KLn) by studying the edge-disjoint paths in KLn with edge faults, smλ(L−1)r(KLn)=(L−1)Lr(n−r)−(L−1)n, where 0≤r≤n−2 and n≥3.