  Since strong quantum fluctuations are essential for the emergence of quantum
spin liquids, there have been extensive exploration and identification of spin
liquid candidates in spin-$1/2$ systems, while such activities are rare in
higher spin systems. Here we report an example of non-Abelian chiral spin
liquid emerging in spin-$3/2$ Heisenberg model on a square lattice. By tuning
Heisenberg exchange interaction and scalar chirality interaction, we map out a
quantum phase diagram enclosing three conventional magnetic orders and a chiral
spin liquid based on density matrix renormalization group studies. The nature
of the spin liquid is identified as a long-sought bosonic version of
Read-Rezayi state that supports non-Abelian Fibonacci anyonic statistics,
identified by the ground state entanglement spectrum. Significantly, we
establish that the non-Abelian CSL emerges through the enlarged local degrees
of freedom and enhanced quantum fluctuations near the classical phase
boundaries of competing magnetic orders. Our numerical discovery of an exotic
quantum spin liquid in spin-$3/2$ system suggests a new route for discovering
fractionalized quantum phases in frustrated higher spin magnetic compounds.
