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.

A small amendment of the result of Hidehiko mentioned in the answer of Moishe Kohan: 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 $X$. It is clear that every path-connected topological space is continuum-connected (but not vice-versa).

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.