Is ionisation an equilibrium process?
An analogy might help. Solubilization of a salt in a solvent (say NaCl in water) is similar to condensation of a gas.
Imagine you have an amount of gas in a cylinder with a piston to regulate the volume and pressure. Beginning at a pressure below the vapor pressure of the substance (which means that all is gas, none is liquid), the pressure is increased (while temperature is constant) until eventually the boiling point is encountered. At this point the pressure is equal to the vapor pressure (at that temperature) and the gas and liquid can coexist (all of this by the way is simpler to illustrate with a T-p phase diagram). Further compression condenses more gas until all of the substance is liquid. Below the boiling point pressure none of the substance is liquid, and above that pressure none is gas.
Next consider solubilization of a substance in a fixed volume of liquid (with T,p constant). One could traverse the solubility diagram by changing the temperature, but it is for the present discussion simpler to consider changing the concentration. At very low concentrations (small amounts of substance) all of the substance solubilizes. Exactly  none (yes, none) is present as the solid. But as you increase the amount of added substance a concentration is reached at which solid remains. That exact point is the solubility limit of the substance at that temperature. It can be thought of as equivalent to the boiling point pressure (coexistence point) of a gas and liquid at a given temperature. If you increase the amount of added substance at that point, no more substance will solubilize (exactly none).
So when a salt is added below its solubility concentration limit no solid remains, all of the substance is in the liquid, and there is no equilibrium only in the sense that there cannot be solid below the solubility limit.

This is also why the equilibrium equation for solubilization of a substance is written as
$$K_{\mathrm{sp}} = \frac{a(\textrm{soln})}{a(\textrm{s})}= a(\textrm{soln})$$
where a's are activities (for small concentrations, the activity of the solubilized substance is equal to concentration ). We ignore the activity of the solid (it is constant and assumed equal to 1) and above the solubility concentration limit the activity of the solubilized substance is also constant (equal to the solubility product).