Theorem 2.2 in this paper implies that every continuum-connected subgroup of $\mathbb R^n$ is a (closed) linear subspace of $\mathbb R^n$.
A topological space $X$ is defined to be continuum-connected if any points of $X$ are contained in a compact connected subset of $X$.
Therefore, the answer to the problem depends on the type of connectedness: for usual connectedness the answer is negative and for continuum-connectedness it is affirmative.