We have selected a sample of local E+A galaxies from the Sloan Digital Sky Survey (SDSS) Data Release 7 for follow-up integral field spectroscopy with the Wide Field Spectrograph (WiFeS) on the Australian National University's (ANU) 2.3-m telescope. The sample was selected using the Hα line in place of the [O ii] λ3727 line as the indicator of ongoing star formation (or lack thereof). This allowed us to select a lower redshift sample of galaxies than available in the literature since the [O ii] λ3727 falls off the blue end of the wavelength coverage in the SDSS for the very lowest redshift objects. This low-redshift selection means that the galaxies have a large angular to physical scale which allows us to resolve the central ∼1 kpc region of the galaxies; the region where stellar population gradients are expected. Such observations have been difficult to make using other higher redshift samples because even at redshifts z∼ 0.1 the angular to physical scale is similar to the resolution provided by ground-based seeing. Our integral field spectroscopy has enabled us to make the first robust detections of Balmer line gradients in the centres of E+A galaxies. Six out of our sample of seven, and all the galaxies with regular morphologies, are observed to have compact and centrally concentrated Balmer line absorption. This is evidence for compact young cores and stellar population gradients which are predicted from models of mergers and tidal interactions which funnel gas into the galaxy core. Given the generally isolated nature of our sample, this argues for the galaxies being seen in the late stage of a merger where the progenitors have already coalesced.