A small amendment of the result of Hidehiko mentioned in the answer of Moishe Kohan: Theorem 2.2 in [this paper][1] 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. 


  [1]: https://projecteuclid.org/journals/topological-methods-in-nonlinear-analysis/volume-54/issue-1/The-continuity-of-additive-and-convex-functions-which-are-upper/10.12775/TMNA.2019.040.full