The "$GL_2(\mathbb{Z}[1/p])$-part" of an element in $GL_2(\mathbb{Q}_p)$

Question

Is there a group homomorphism $\varphi:GL_2(\mathbb{Q}_p)\rightarrow GL_2(\mathbb{Z}[1/p])$ such that for every $x\in GL_2(\mathbb{Q}_p)$ we have $x^{-1}\cdot\varphi(x)\in GL_2(\mathbb{Z}_p)$?

My motivation is that it is true in the "$GL_1$ case" (with $x\mapsto p^{v_p(x)}$).