What do you mean by "nontrivial". I would call a subspace nontrivial as long as it was not just the 0 vector. If that's the case then just take any vectors space V with V as the subspace.
Another possibility, a lot more non-trivial would be an infinite dimensional vector space with an infinite dimensional subspace.
The vector space of all polynomials with subspace the space of all even polynomials comes to mind.
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