We introduce the notion of Hilbert $C^*$-module independence: Let $\\\\mathscr{A}$ be a unital $C^*$-algebra and let $\\\\mathscr{E}_i\\\\subseteq \\\\mathscr{E},\\\\,\\\\,i=1, 2$, be ternary subspaces of a Hilbert $\\\\mathscr{A}$-module $\\\\mathscr{E}$. Then $\\\\mathscr{E}_1$ and $\\\\mathscr{E}_2$ are said to be Hilbert $C^*$-module independent if there are positive constants $m$ and $M$ such that for every state $\\\\varphi_i$ on $\\\\langle \\\\mathscr{E}_i,\\\\mathscr{E}_i\\\\rangle,\\\\,\\\\,i=1, 2$, there exists a state $\\\\varphi$ on $\\\\mathscr{A}$ such that \\\\begin{align*} m\\\\varphi_i(|x|)\\\\leq \\\\varphi(|x|) \\\\leq M\\\\varphi_i(|x|^2)^{\\\\frac{1}{2}},\\\\qquad \\\\mbox{for all~}x\\\\in \\\\mathscr{E}_i, i=1, 2. \\\\end{align*} We show that it is a natural generalization of the notion of $C^*$-independence of $C^*$-algebras. Moreover, we demonstrate that even in the case of $C^*$-algebras this concept of independence is new and has a nice characterization in terms of Hahn--Banach type extensions. We show that if $\\\\langle \\\\mathscr{E}_1,\\\\mathscr{E}_1\\\\rangle $ has the quasi extension property and $z\\\\in \\\\mathscr{E}_1\\\\cap \\\\mathscr{E}_2$ with $\\\\|z\\\\|=1$, then $|z|=1$. Several characterizations of Hilbert $C^*$-module independence and a new characterization of $C^*$-independence are given. One of characterizations states that if $z_0\\\\in \\\\mathscr{E}_1\\\\cap \\\\mathscr{E}_2$ is such that $\\\\langle z_0,z_0\\\\rangle=1$, then $\\\\mathscr{E}_1$ and $\\\\mathscr{E}_2$ are Hilbert $C^*$-module independent if and only if $\\\\|\\\\langle x,z_0\\\\rangle\\\\langle y,z_0\\\\rangle\\\\|=\\\\|\\\\langle x,z_0\\\\rangle\\\\|\\\\,\\\\|\\\\langle y,z_0\\\\rangle\\\\|$ for all $x\\\\in \\\\mathscr{E}_1$ and $y\\\\in \\\\mathscr{E}_2$. We also provide some technical examples and counterexamples to illustrate our results.

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