In the atomic force microscopy (AFM) investigation of soft polymers and liquids, the tip-sample interaction is dominated by long-range van der Waals forces, capillary forces and adhesion. Furthermore, the tip can indent several tens of nanometres into the surface, and it can pull off a polymer filament from the surface. Therefore, measuring the unperturbed shape of a polymeric fluid can be challenging. Here, we study the tip-sample interaction with polystyrene droplets swollen in chloroform vapour, where we can utilize the solvent vapour concentration to adjust the specimen's mechanical properties from a stiff solid to a fluid film. With the same AFM tip, we use two different AFM force spectroscopy methods to measure three-dimensional (3D) depth profiles of the tip-sample interaction: force-distance (FD) curves and amplitude-phase-distance (APD) curves. The 3D depth profiles reconstructed from FD and APD measurements provide detailed insight into the tip-sample interaction mechanism for a fluid polymer solution. The fluid's intrinsic relaxation time, which we measure with an AFM-based step-strain experiment, is essential for understanding the tip-sample interaction mechanism. Furthermore, measuring 3D depth profiles and using APD data to reconstruct the unperturbed surface comprise a versatile methodology for obtaining accurate dimensional measurements of fluid and gel-like objects on the nanometre scale.