In the context of star formation through fragmentation of an extremely metal-deficient protogalactic cloud, the gravitational collapse of filamentary gas clouds is explored with 1D numerical hydrodynamics coupled with non-equilibrium chemistry of H$_2$ and HD. It is found that the cloud evolution is governed mainly by the initial central density ($n_{\rm c,0}$) and H$_2$ abundance ($x_{\rm H_2,0}$). In particular, the evolution of low-density filaments bifurcates at a threshold H$_2$ abundance of $x_{\rm H_2,cr}\simeq 3 \times 10^{-3}$, beyond which HD cooling overwhelms H$_2$ cooling. The contraction of a low density filament with $x_{\rm H_2, 0}\gtrsim x_{\rm H_2,cr}$ is strongly decelerated when $n_{\rm c}$ reaches a critical density of HD, and the filament is expected to fragment at $\sim 10^{7}$ cm$^{-3}$. The fragment mass is lowered to be $\approx 10M_\odot$. In contrast, the contraction of a low density filament with $x_{\rm H_2, 0}\lesssim x_{\rm H_2,cr}$ is regulated by H$_2$ cooling. In this case, the fragment mass is as high as $\approx 10^2M_\odot$. For a high-density filament, the cloud evolution is governed by H$_2$ cooling. The fragmentation is not expected to take place until the cloud becomes opaque in H$_2$ lines at $n_{\rm c,0}\sim 10^{12-13}$ cm$^{-3}$, so that the fragment mass is reduced to 1-2 M$_\odot$. As a result, the stellar IMF could be bimodal and deficient in sub-solar mass stars, where the high mass peak is around $10M_\odot$ or $10^2M_\odot$, dependently on $n_{\rm c,0}$ and $x_{\rm H_2,0}$.

10 pages, 6 figures, accepted by ApJ