Yes, it is a K-32 -- rare model and highly sought by collectors.
K222xxx would have shipped in 1954 or 1955. It should not have a model number on it, because those were introduced in 1957. In that year the K-32 Masterpiece became the Model 16.
Your gun should also be a five-screw model, which means four screws on the sideplate. One would be concealed under the stocks, so the easiest way to tell if you have a five-screw gun is to look at the top of the sideplate in front of the hammer. If there is a screw there, you have a five-screw model. The fifth screw was engineered away in 1955, at which time the affected revolvers became known as four-screw models. When counting screws, you also include the one on the frame in front of the trigger guard. When that one disappeared a few years after model numbers were introduced, Smiths became three-screw guns.
Pictures are definitely warranted. Congratulations on owning a revolver that is a grail gun for many. And welcome to the forum.