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Is there an example of a module $M$ over a commutative Noetherian local ring $(R,\mathfrak m, k)$ such that $\text{Ext}_R^{>0}(k, M)=0$ but $M$ is not an injective $R$-module?

I know that for such an example, $R$ can not be Artinian, nor can $M$ be finitely generated ...

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If you take $R=k[[x,y]]$ and $M=k[[x]][x^{-1}]$, this should give you an example. $M$ is not injective because there is a non-split injective map $M \to R[x^{-1},y^{-1}]/yR[x^{-1}]$. But if you compute Ext using the Koszul complex on $x$ and $y$ to resolve $k$ over $R$, you find that $\operatorname{\rm Ext}^*_R(k,M)=0$. Alternatively, multiplication by $x$ is an isomorphism on $M$ but zero on $k$, so both an isomorphism and zero on $\operatorname{\rm Ext}^*_R(k,M)=0$.

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