  We report on magnetization, sound velocity, and magnetocaloric-effect
measurements of the Ising-like spin-1/2 antiferromagnetic chain system
BaCo$_2$V$_2$O$_8$ as a function of temperature down to 1.3 K and applied
transverse magnetic field up to 60 T. While across the N\'{e}el temperature of
$T_N\sim5$ 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 $B^{c,3D}_\perp=21.4$ T,
indicating the suppression of the antiferromagnetic order. At higher fields,
the $T(B)$ curve shows a broad minimum at $B^c_\perp = 40$ T, accompanied by a
broad minimum in the sound velocity and a saturation-like magnetization. These
features signal a quantum phase transition which is further characterized by
the divergent behavior of the Gr\"{u}neisen parameter $\Gamma_B \propto
(B-B^{c}_\perp)^{-1}$. By contrast, around the critical field, the
Gr\"{u}neisen parameter converges as temperature decreases, pointing to a
quantum critical point of the one-dimensional transverse-field Ising model.
