Let $X$ and $Y$ be compact metric spaces and let $\mathcal{M}_n(\mathbb{C})$ be the Banach algebra of all $n\times n$ complex matrices. We prove that the set of all unital surjective linear isometries from $\mathrm{Lip}(X,\mathcal{M}_n(\mathbb{C}))$ to $\mathrm{Lip}(Y,\mathcal{M}_n(\mathbb{C}))$, whenever both spaces are endowed with the sum norm, is topologically reflexive.