Does the rate constant depend on pressure?
According to hard-sphere collision theory of gas-phase reactions, the rate constant ($k$) for the elementary bimolecular reaction $\ce{B + C -&gt; products}$ is
$$ k = N_\text{A} \pi (r_\text{B} + r_\text{C})^2 \left[ \frac{8RT}{\pi}\left( \frac{1}{M_\text{B}} + \frac{1}{M_\text{C}}\right)\right]^{1/2}e^{-E_\text{thr}/RT}$$
where $N_\text{A}$ is the Avogadro constant, $r_\text{B}$ and $r_\text{C}$ are the radii of the hard-spheres, $R$ is the ideal gas constant, $M_\text{B}$ and $M_\text{C}$ are the molar masses of B and C, and $E_\text{thr}$ is the threshold energy on a per-mole basis. Therefore, no pressure dependence.

For nonideal systems, the rate law for the same elementary reaction can be written as
$$ r = -\frac{d[\text{B}]}{dt} = -\frac{d[\text{C}]}{dt} = k_r [\text{B}][\text{C}]$$
The Brønsted–Bjerrum equation relates the apparent rate constant, $k_r$ with the infinite-dilution rate constant, $k_r^\infty$, and the activity coefficients of the envolved species:
$$  k_r = \left( \frac{\gamma_\text{B}\gamma_\text{C}}{\gamma^\ddagger}\right)k_r^\infty$$
where $\gamma_\text{B}$, $\gamma_\text{C}$, and $\gamma^\ddagger$ are the concentration-scale activity coefficients of B, C, and the transition state respectively. These quantities depend on temperature, pressure, and on the concentration of all species on the same phase. For this reason, the rate constant of nonideal elementary reactions can have a dependence on the pressure.