Mader [J. Combin. Theory Ser. B 40 (1986) 152-158] proved that every $k$-edge-connected graph $G$ with minimum degree at least $k+1$ contains a vertex $u$ such that $G-\{u\}$ is still $k$-edge-connected. In this paper, we prove that every $k$-edge-connected graph $G$ with minimum degree at least $k+2$ contains an edge $uv$ such that $G-\{u,v\}$ is $k$-edge-connected for any positive integer $k$. In addition, we show that for any tree $T$ of order $m$, every $k$-edge-connected graph $G$ with minimum degree greater than $4(k+m)^2$ contains a subtree $T'$ isomorphic to $T$ such that $G-V(T')$ is $k$-edge-connected.