If $V$ is a finite dimensional $K$-vector space, then every set of subsets of subspaces contains a maximal element, i.e. a subspace which no subspace of the set contains properly, equivalently we have no infinitely ascending chain of subspaces, this is an easy consequence of finite-dimensionality.
For a vector space $V$, not every subgroup of $(V, +)$ is also a subspace, for example $\mathbb R$ considered as a vector space over itself has only the trivial subspaces, but among the subgroups are $\mathbb Q$ or $\mathbb Z$.
But is it possible that a subspace contains an infinite ascending (or infinite descending) sequence of subgroups? I am asking for a finite dimensional vector space $V$ with subspace $U \le V$, such that we have an infinitely ascending chain $$ A_1 < A_2 < A_3 < \ldots $$ of subgroups $(A_i, +) \le (V, +)$ of the additive group of $V$, but $A_i < U$ for all $i$.