We present Integral Field Unit GMOS-IFU data of the compact H II galaxy UM 408, obtained at the Gemini South telescope, in order to derive the spatial distribution of emission lines and line ratios, kinematics, plasma parameters, and oxygen abundances as well the integrated properties over an area of 3''×4farcs4 equivalent with ~750 pc × 1100 pc located in the central part of the galaxy. The starburst in this area is resolved into two giant regions of about 1farcs5 and 1'' (~375 and ~250 pc) diameter, respectively and separated 1.5-2'' (~500 pc). The extinction distribution concentrate its highest values close but not coincident with the maxima of Hα emission around each one of the detected regions. This indicates that the dust has been displaced from the exciting clusters by the action of their stellar winds. The ages of these two regions, estimated using Hβ equivalent widths, suggest that they are coeval events of ~5 Myr with stellar masses of ~104 M sun. We have also used [O III]/Hβ and [S II]/Hα ratio maps to explore the excitation mechanisms in this galaxy. Comparing the data points with theoretical diagnostic models, we found that all of them are consistent with excitation by photoionization by massive stars. The Hα emission line was used to measure the radial velocity and velocity dispersion. The heliocentric radial velocity shows an apparent systemic motion where the east part of the galaxy is blueshifted, while the west part is redshifted, with a relative motion of ~10 km s-1. The velocity dispersion map shows supersonic values typical for extragalactic H II regions. We derived an integrated oxygen abundance of 12+log(O/H) = 7.87 summing over all spaxels in our field of view. An average value of 12+log(O/H) = 7.77 and a difference of Δ(O/H) = 0.47 between the minimum and maximum values (7.58 ± 0.06-8.05 ± 0.04) were found, considering all data points where the oxygen abundance was measured. The spatial distribution of oxygen abundance does not show any significant gradient across the galaxy. On the other hand, the bulk of data points are lying in a region of ±2σ dispersion (with σ = 0.1 dex) around the average value, confirming that this compact H II galaxy, like other previously studied dwarf irregular galaxies, is chemically homogeneous. Therefore, the new metals processed and injected by the current star formation episode are possibly not observed and reside in the hot gas phase, whereas the metals from previous events are well mixed and homogeneously distributed through the whole extent of the galaxy.