We study the evidence for spin liquid in the frustrated diamond lattice antiferromagnet CoAl₂O₄ by means of single-crystal neutron scattering in zero and applied magnetic fields. The magnetically ordered phase appearing below T_N=8 K remains nonconventional down to 1.5 K. The magnetic Bragg peaks at the q=0 positions are broad and their line shapes have strong Lorentzian contributions. Additionally, the peaks are connected by weak diffuse streaks oriented along the 111 directions. The observed short-range magnetic correlations are explained within the spiral spin-liquid model. The specific shape of the energy landscape of the system, with an extremely flat energy minimum around q=0 and many low-lying excited spiral states with q=111, results in thermal population of this manifold at finite temperatures. The agreement between the experimental results and the spiral spin-liquid model is only qualitative, indicating that microstructure effects might be important to achieve quantitative agreement. Application of a magnetic field significantly perturbs the spiral spin-liquid correlations. The magnetic peaks remain broad but acquire more Gaussian line shapes and increase in intensity. The 1.5 K static magnetic moment increases from 1.58 μ_B/Co at zero field to 2.08 μ_B/Co at 10 T. The magnetic excitations appear rather conventional at zero field. Analysis using classical spin-wave theory yields values of the nearest- And next-nearest-neighbor exchange parameters J ₁=0.92(1) meV and J₂=0.101(2) meV and an additional anisotropy term D=-0.0089(2) meV for CoAl₂O₄. In the presence of a magnetic field, the spin excitations broaden considerably and become nearly featureless at the zone center.