Yes. Pressure, as a concept, is simply the force exerted by random molecular motion per unit area, and since all matter is composed of molecules, and all matter above absolute zero has some random motion, everything has molecules exerting some force over some area. (This pressure is the vapor pressure of the solid, which is typically extremely low. If the external pressure on the solid is higher than the vapor pressure of the solid, the solid molecules will not fly off and escape. Things with high vapor pressure exert a high "pressure" on their surroundings and thus escape easily - jet fuel or nail polish remover have very high vapor pressures.)
If you have a liquid reaction between molecules A and B, bringing the liquid to a higher pressure will increase the frequency of collisions between molecules of A and B (by the definition of pressure). Since there is a finite probability of the reaction proceeding for any given collision, increasing the collisions increases the rates.
Practically, for you to see that kind of effect in a laboratory would require a very pressure-sensitive reaction, large molecules whose shape changes with pressure, or a reaction with a large change in the number of moles. Biological chemistry is rich with examples pressure-sensitive non-gaseous reactions, mainly due to pressure changing the shape and behavior of large complex molecules.
There's also the effect of osmotic pressure, which is a pressure induced not by random molecular motion but by the tendency of the substance to seek a lower free energy state. Osmotic pressure will affect concentrations (the system will seek to reduce the concentrations of ions), which will affect reaction rates.
Increasing the pressure of solids and liquids can also change their physical properties (heat capacity and thermal conductivity are two excellent examples). These physical properties are innately tied to the reaction mechanism. Since most reactions occur in sequence, if the physical properties are different (and if they change over the course of the reaction), pressure will exert an effect there.
Finally, there is an effect technically excluded by your question, but important nonetheless, and that is the effect of pressure on solid-gas reactions (sublimation or deposition). The manufacture of integrated circuits involves depositing materials onto chips in layers, and pressure is an important way to control that rate.
In general, the more nonlinear the reaction mechanism, the more likely it is that the small effects that pressure creates will cascade into larger changes that are what you would call "significant."