Problem
Find all vector spaces that have exactly one basis
Solution
The only vector space that has exactly one basis is $\lbrace 0 \rbrace$, proof:
Let $v_1, v_2, \dots, v_n$ be a basis of $V$, then for all non-zero $\lambda$ $\lambda v_1, \lambda v_2, \dots, \lambda v_n$ is also a basis as per P2A8. Therefore only the empty list basis can be a unique basis, the empty list is a basis for $\lbrace 0 \rbrace$.