  We report a study of NiBi$_3$ single crystals by means of
electrical-resistivity-, magnetization-, and muon-spin rotation and relaxation
($\mu$SR) measurements. As a single crystal, NiBi$_3$ adopts a needle-like
shape and exhibits bulk superconductivity with $T_c \approx 4.1$ K. By applying
magnetic fields parallel and perpendicular to the $b$-axis of NiBi$_3$, we
establish that its lower- and upper critical fields, as well as the magnetic
penetration depths show slightly different values, suggesting a weakly
anisotropic superconductivity. In both cases, the zero-temperature upper
critical fields are much smaller than the Pauli-limit value, indicating that
the superconducting state is constrained by the orbital pair breaking. The
temperature evolution of the superfluid density, obtained from transverse-field
$\mu$SR, reveals a fully-gapped superconductivity in NiBi$_3$, with a shared
superconducting gap $\Delta_0$ = 2.1 $k_\mathrm{B}$$T_c$ and magnetic
penetration depths $\lambda_0$ = 223 and 210 nm for $H \parallel b$- and $H
\perp b$, respectively. The lack of spontaneous fields below $T_c$ indicates
that time-reversal symmetry is preserved in NiBi$_3$. The absence of a fast
muon-spin relaxation and/or precession in the zero-field $\mu$SR spectra
definitely rules out any type of magnetic ordering in NiBi$_3$ single crystals.
Overall, our investigation suggests that NiBi$_3$ behaves as a conventional
$s$-type superconductor.
