Let $V$ be a countably infinite dimensional $K$-vector space over a local field $K$ (nontrivially discretely valued with finite residue field). Let $o$ be the ring of integers of $K$.

Given two free o-submodules $L_1,L_2$ in $V$, does $L_1+L_2 \subsetneqq V$ hold? (Here L_1+L_2 denotes the smallest $o$-module inside $V$ containing $L_1$ and $L_2$)