Summary: Let \((X,\varrho,\mu)_{d,\theta}\) be a space of homogeneous type, i.e., \(X\) is a set, \(\varrho\) is a quasi-metric on \(X\) with the property that there are constants \(\theta\in (0,1]\) and \(C_0>0\) such that for all \(x, x', y \in X\), \[ | \varrho(x,y)-\varrho(x',y)| \leq C_0\varrho(x,x')^\theta[\varrho(x,y) + \varrho(x',y)]^{1-\theta}, \] and \(\mu\) is a nonnegative Borel regular measure on \(X\) such that for some \(d>0\) and all \(x\in X\), \[ \mu(\{y\in X: \varrho(x,y)<r\})\sim r^d. \] Let \(\varepsilon\in (0,\theta]\), \(|s| <\varepsilon\) and \(\max\{d/(d+\varepsilon),d/(d+s+\varepsilon)\} < q \leq \infty.\) The author introduces new inhomogeneous Triebel–Lizorkin spaces \({F^s_{\infty q}(X)}\) and establishes their frame characterizations by first establishing a Plancherel-Pólya-type inequality related to the norm \(\| \cdot\|_{F^s_{\infty q}(X)}\), which completes the theory of function spaces on spaces of homogeneous type. Moreover, the author establishes the connection between the space \({F^s_{\infty q}(X)}\) and the homogeneous Triebel-Lizorkin space \({\dot F^s_{\infty q}(X)}\). In particular, he proves that bmo\((X)\) coincides with \(F^0_{\infty 2}(X)\).