Sum of all vectors in a subspace of a vector space over a finite field

Question

Let $F_n$ be a finite field where $n$ is a power of $2$.

Let $F_n^k$ be the set of all vectors of length $k$ over $F_n$, where $k \geq 2$.

Now, I know that the sum of all the vectors in $F_n^k$ is equal to the zero vector.

My question is the following;

What if we take a subspace of $F_n^k$, where the dimension of the subspace is greater than $1$? Is the sum still the zero vector?

My thoughts are that as this subspace is a vector space in its own right, then the same result should hold? I’m just unsure!