We present experimental evidence for a quantum phase transition in the easy-axis S=3/2 anisotropic quasi-one-dimensional ferromagnet CoCl2⋅2D2O in a transverse field. Elastic neutron scattering shows that the magnetic order parameter vanishes at a transverse critical field μ0Hc=16.05(4) T, while inelastic neutron scattering shows that the gap in the magnetic excitation spectrum vanishes at the same field value, and reopens for H>Hc. The field dependence of the order parameter and the gap are well described by critical exponents β=0.45±0.09 and zν close to 1/2, implying that the quantum phase transition in CoCl2⋅2D2O differs significantly from the textbook version of a S=1/2 Ising chain in a transverse field. We attribute the difference to weak but finite three-dimensionality of the magnetic interactions.