In a graph $G$, an {\it efficient dominating set} is a subset $D$ of vertices such that $D$ is an independent set and each vertex outside $D$ has exactly one neighbor in $D$. The {\textsc{Efficient Dominating Set}} problem (EDS) asks for the existence of an efficient dominating set in a given graph $G$. The EDS is known to be $NP$-complete for $P_7$-free graphs, and is known to be polynomial time solvable for $P_5$-free graphs. However, the computational complexity of the EDS problem is unknown for $P_6$-free graphs. In this paper, we show that the EDS problem can be solved in polynomial time for a subclass of $P_6$-free graphs, namely ($P_6$, banner)-free graphs.