We report on magnetization, sound-velocity, and magnetocaloric-effect measurements of the Ising-like spin-1/2 antiferromagnetic chain system BaCo2V2O8 as a function of temperature down to 1.3 K and an applied transverse magnetic field up to 60 T. While across the Néel temperature of TN∼5 K anomalies in magnetization and sound velocity confirm the antiferromagnetic ordering transition, at the lowest temperature the field-dependent measurements reveal a sharp softening of sound velocity v(B) and a clear minimum of temperature T(B) at Bc,3D=21.4 T, indicating the suppression of the antiferromagnetic order. At higher fields, the T(B) curve shows a broad minimum at Bc=40 T, accompanied by a broad minimum in the sound velocity and a saturationlike magnetization. These features signal a quantum phase transition, which is further characterized by the divergent behavior of the Grüneisen parameter ΓB (B-Bc)-1. By contrast, around the critical field, the Grüneisen parameter converges as temperature decreases, pointing to a quantum critical point of the one-dimensional transverse-field Ising model.