In this letter, we develop a continuous fluid antenna (FA) framework for uplink channel estimation in cell-free massive multiple-input and multiple-output (CF-mMIMO) systems. By modeling the wireless channel as a spatially correlated Gaussian random field, channel estimation is formulated as a Gaussian process (GP) regression problem with motion-constrained spatial sampling. Closed-form expressions for the linear minimum mean squared error (LMMSE) estimator and the corresponding estimation error are derived. A fundamental comparison with discrete port-based architectures is established under identical position constraints, showing that continuous FA sampling achieves equal or lower estimation error for any finite pilot budget, with strict improvement for non-degenerate spatial correlation models. Numerical results validate the analysis and show the performance gains of continuous FA sampling over discrete baselines.