  Motivated by the mathematical beauty and the recent experimental realizations
of fractal systems, we study the spin-$1/2$ antiferromagnetic Heisenberg model
on a Sierpi\'nski gasket. The fractal porous feature generates new kinds of
frustration to exhibit exotic quantum states. Using advanced tensor network
techniques, we identify a quantum gapless-spin-liquid ground state in
fractional spatial dimension. This fractal spin system also demonstrates
nontrivial non-local properties. While the extremely short-range correlation
causes a highly degenerate spin form factor, the entanglement in this fractal
system suggests scaling behaviors significantly different from those in integer
dimensions. We also study the dynamic structure factor and clearly identify the
gapless excitation with a stable corner excitation emerged from the
ground-state entanglement. Our results unambiguously point out multiple
essential properties of this fractal spin system, and open a new route to
explore spin liquid and frustrated magnetism.
