We investigate the properties of an impurity immersed in an ensemble of spin-polarized fermions confined in a tight quantum wire. We use a non-perturbative variational approach that accounts for virtual transverse excitations and regularizes the zero-range interaction. We compute the polaron’s energy, effective mass, and spectral weight, and benchmark our results against the exactly solvable Yang-Gaudin model of a purely 1D system. While the two models agree in the weakly interacting regime, we find significant deviations in the strongly attractive limit, including a divergence of the effective mass and evidence for a a breakdown of the polaron as the ground state of the system inherited from the underlying three-dimensional physics, absent in the purely 1D description. Our work quantifies the accuracy of quasi-one-dimensional systems as analog quantum simulators and highlights the emergence of beyond-1D physics even in nominally one-dimensional settings, with direct implications for cold-atom experiments and low-dimensional quantum matter.