Context. Blue compact galaxies (BCG) are gas-rich, low-mass, small systems that form stars at unusually high rates. This makes them excellent laboratories for investigating the process of star-formation (SF) at galactic scales and the effects of massive stellar feedback on the interstellar (and intergalactic) medium. Aims. We analyzed the BCG Tololo 1937-423 using optical integral field spectroscopy to probe its morphology, stellar content, nebular excitation and ionization properties, and the kinematics of its warm ionized gas. Methods. Tololo 1937-423 was observed with the Visible Multi-Object Spectrograph at the Very Large Telescope. We took data in the wavelength range 4150-7400 Å, covering a field of view of 27 ′′ × 27 ′′ on the sky with a spatial sampling of 0 ′′ . 67. From these data we built maps in the continuum and brighter emission lines, diagnostic line ratio maps, and velocity dispersion fields. We also generated the integrated spectrum of the main H ii regions and young stellar clusters to determine reliable physical parameters and oxygen abundances. Results. We found that Tololo 1937-423 is currently undergoing an extended starburst. In the Hα maps we identified nine major clumps, aligned mostly northeast-southwest, and stretching to galactocentric distances ≥ 2 kpc. The galaxy presents a single continuum peak that is not cospatial with any knot in emission lines, indicating at least two relatively recent episodes of SF. The inhomogeneous dust distribution reachs its maximum [E(B-V)∼0.97] roughly at the position of the continuum peak. We found shocked regions in the galaxy outer regions and at the edges of the SF knots. The oxygen abundance, 12+log(O/H)∼8.20±0.1, is similar in all the SF regions, suggesting a chemically homogeneous ionized interstellar medium over spatial scales of several kpc. The ionized gas kinematics displays an overall regular rotation around a northwest-southeast axis, with a maximum velocity of 70±7 km s −1 . Conclusions. The morphology of the galaxy and the two different episodes of SF suggest a scenario of triggered (induced by supernova shock waves) SF acting in Tololo 1937-423. The inferred ages for the different SF episodes (∼13-80 Myr for the central post-starburst and 5-7 Myr for the ongoing SF) are consistent with triggered SF, with the most recent SF episode caused by the collective effect of stellar winds and supernova explosions from the central post-starburst. The velocity dispersion pattern, with higher velocity dispersions found at the edges of the SF regions, and shocked regions in the galaxy, also favor this scenario.