Aaron Meyerowitz suggested to try to reduce the problem to trees and, to me, this seems to work. First we can suppose that $G$ is a connected graph, because we can solve the problem separatly for each component. It is easy to see by Zorn's Lemma, that every connected graph contains a spanning tree, i.e. a subgraph which is a tree and which connects all vertices of the original graph. Hence it is enough to solve the problem for a tree.
Put $E_0=\emptyset$. We choose a root $r$ of the tree and denote by $L_n$ the set of vertices which are $n$ edges far from $r$. By hypothesis, $L_1$ is nonempty. If $L_1$ contains at least one vertex of degree 1, we define $E_1$ to be exactly the edges connecting $r$ with the vertices from $L_1$ of degree 1. Otherwise, we pick arbitrary $x_1$ from $L_1$ and define $E_1$ as a singleton consisting just of the edge connecting $r$ and $x_1$. Now we continue inductively by level $n$ of the tree (which is easily well-defined). Let $v \in L_n$, put $E_n=E_{n-1}$:
- If $v$ is leaf, i.e. the tree "under" $v$ has just one vertex, do nothing.
- If there is an edge from $v$ to an element in $L_{n-1}$, add to $E_n$ all edges connecting $v$ with leaves under $v$.
- Otherwise, apply to $v$ the same procudere as to $r$ (if there is a leaf under $v$, add all the edges connecting $v$ with leaves to $E_n$, otherwise pick some edge and add it to $E_n$).
Put $E'=\bigcup E_n$, this (I think) is the desired subset of edges, since:
Let $v$ be a vertex, then $v \in L_n$ for some $n \geq 0$.
$\operatorname{deg}v \geq 1$: Suppose there is no edge connecting $v$ with any edge from level $n-1$. Then by the construction there must be an edge from $v$ to some vertex in level $n+1$.
Suppose $v$ has degree 1. Then by the construction, the parent of $v$ is connected only to vertices of degree $1$. Thus there is no path of edge-wise length more then 3.
Thanks for every comment.