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(This should be a comment)

What about the flat torus $\mathbb{S}^1\times \mathbb{S}^1$? I think you need to amend the question to ask for non-positive sectional curvature.

I should add that by infinite dimensional morse theory (for the energy functional on the loop space of $M$-which satisfies the Palais-Smale condition) you should (in principal) be able to conclude that each component of the loop space is contractible. In other words the homotopy groups vanish for $k>1$ that is, $\pi_k(M)=0$ for all $k>1$.
I'm not sure if that is enough to ensure the existence of a non-positively curved metric on $M$ but is certainly suggestive...
What about the flat torus $\mathbb{S}^1\times \mathbb{S}^1$? I think you need to amend the question to ask for non-positive sectional curvature.