Context. Blue compact galaxies (BCG) are narrow emission-line systems that undergo a violent burst of star formation. They are compact, low-luminosity galaxies, with blue colors and low chemical abundances, which offer us a unique opportunity to investigate collective star formation and its effects on galaxy evolution in a relatively simple, dynamically unperturbed environment. Spatially resolved spectrophotometric studies of BCGs are essential for a better understanding of the role of starburst-driven feedback processes on the kinematical and chemical evolution of low-mass galaxies near and far. Aims: We carry out an integral field spectroscopy (IFS) study of a sample of luminous BCGs, with the aim to probe the morphology, kinematics, dust extinction, and excitation mechanisms of their warm interstellar medium (ISM). Methods: We obtained IFS data for five luminous BCGs with VIRUS-P, the prototype instrument for the Visible Integral Field Replicable Unit Spectrograph, attached to the 2.7 m Harlan J. Smith Telescope at the McDonald Observatory. VIRUS-P consists of a square array of 247 optical fibers, which covers a 109″ × 109″ field of view, with a spatial sampling of 4farcs2 and a 0.3 filling factor. We observed in the 3550-5850 Å spectral range, with a resolution of 5 Å FWHM. From these data we built two-dimensional maps of the continuum and the most prominent emission-lines ([O ii] λ3727, Hγ, Hβ and [O iii] λ5007), and investigated the morphology of diagnostic emission-line ratios and the extinction patterns in the ISM as well as stellar and gas kinematics. Additionally, from integrated spectra we inferred total line fluxes and luminosity-weighted extinction coefficients and gas-phase metallicities. Results: All galaxies exhibit an overall regular morphology in the stellar continuum, while their warm ISM morphology is more complex: in II Zw 33 and Mrk 314, the star-forming regions are aligned along a chain-structure; Haro 1, NGC 4670 and III Zw 102 display several salient features, such as extended gaseous filaments and bubbles. A significant intrinsic absorption by dust is present in all galaxies, the most extreme case being III Zw 102. Our data reveal a plethora of kinematical patterns, from overall regular gas and stellar rotation to complex velocity fields produced by structurally and kinematically distinct components.