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Let $f$ and $g$ be holomorphic or Maass cusp forms for $\rm SL_2(\mathbb{Z})$and let $\chi$ be a primitive Dirichlet character of prime power conductor$\mathfrak{q}=p^{\kappa}$ with $p$ prime and $\kappa>12$. A subconvex bound forthe central values of the Rankin-Selberg $L$-functions $L(s,f\otimes g \otimes\chi)$ is proved in the depth-aspect $$ L\left(\frac{1}{2},f\otimes g \otimes\chi\right)\ll_{f,g,\varepsilon} p^{3/4}\mathfrak{q}^{15/16+\varepsilon}. $$

Bounds for $\rm GL_2\times GL_2$ $L$-functions in depth aspect