  In all known fermionic superfluids, Cooper pairs are composed of spin-1/2
quasi-particles that pair to form either spin-singlet or spin-triplet bound
states. The "spin" of a Bloch electron, however, is fixed by the symmetries of
the crystal and the atomic orbitals from which it is derived, and in some cases
can behave as if it were a spin-3/2 particle. The superconducting state of such
a system allows pairing beyond spin-triplet, with higher spin quasi-particles
combining to form quintet or septet pairs. Here, we report evidence of
unconventional superconductivity emerging from a spin-3/2 quasiparticle
electronic structure in the half-Heusler semimetal YPtBi, a low-carrier density
noncentrosymmetric cubic material with a high symmetry that preserves the
$p$-like $j=3/2$ manifold in the Bi-based $\Gamma_8$ band in the presence of
strong spin-orbit coupling. With a striking linear temperature dependence of
the London penetration depth, the existence of line nodes in the
superconducting order parameter $\Delta$ is directly explained by a
mixed-parity Cooper pairing model with high total angular momentum, consistent
with a high-spin fermionic superfluid state. We propose a $\mathbf{k\cdot p}$
model of the $j=3/2$ fermions to explain how a dominant $J$=3 septet pairing
state is the simplest solution that naturally produces nodes in the mixed
even-odd parity gap. Together with the underlying topologically non-trivial
band structure, the unconventional pairing in this system represents a truly
novel form of superfluidity that has strong potential for leading the
development of a new generation of topological superconductors.
