We first evaluate the thermal model. If we assume a representative intrinsic dimension of 1.9 0.9 0.9 kpc (i.e. 2 1 1 ), as suggested for the bright ``core'' by the X-ray image (Fig. 3 and Fig. 4), the required gas density for the best fitting Raymond-Smith model is n 3 cm. Because the X-rays come from the same general region of space as the radio and optical emission (Fig. 3), we may estimate the magnetic field in this putative thermal gas from the synchrotron emission. For an equipartition magnetic field of 4.7 10 gauss (obtained assuming a total cosmic ray energy equal to twice that of the electrons, = 0.74 [Meisenheimer, Yates & Röser 1997], lower and upper cut-off frequencies 10 and 10 Hz, respectively, and an emitting volume corresponding to the size of the radio core - 760 190 190 pc [0.''8 0.''2 0.''2]), the rotation measure through the hot spot is expected to be 6 10 rad m. If the larger volume adopted above for the X-ray source is used to calculate the equipartition field, a value of 1.4 10 gauss and a rotation measure through the hot spot of 2 10 rad m are found. In contrast, the high polarization observed at 6 cm implies a rotation measure internal to the hot spot 900 rad m (n 0.014 cm), a factor of 200 lower than required for thermal emission. If the 20 cm polarization is used, the upper limit on rotation measure and density are an order of magnitude smaller, but the hot spot is not well resolved at this wavelength. We thus conclude that our upper limit to the gas density renders a thermal model untenable. This conclusion could be wrong if either a) the magnetic field has a preferred direction but many reversals; such a field structure can provide the high observed synchrotron polarization but gives little Faraday rotation, or b) the thermal gas is actually in small, dense clumps with a small covering factor. We favor neither of these: the field structure of a) is physically implausible, and it is very improbable that thermal gas of density 3 cm, let alone the higher density required if the gas is clumped, exists 240 kpc from the Pictor A galaxy. Another argument for a low gas density in the hot spot comes from the photoelectrically absorbing column. Most or all of this column comes from gas in our Galaxy; any contribution from other regions is 3 10 atoms cm (Tab. 1). If spread uniformly throughout the hot spot ``core'' of average diameter 1 kpc, the density is 0.1 cm (as long as electrons are not stripped from the K shells of the relevant elements), a factor of 30 below the density needed for thermal emission. In summary, our finding (Section 3.2.2) that a thermal model provides a poor description of the X-ray spectrum, and the upper limits to the gas density from the limits on Faraday depolarization and the absorbing column, rule out a thermal model of the X-rays from the hot spot We conclude that the X-rays are non-thermal in origin and discuss relevant models in Section 4.2.2.