We present near-infrared (NIR) Very Large Telescope (VLT) Infrared Spectrometer and Array Camera (ISAAC) imaging and spectroscopy of the peculiar Virgo galaxy NGC4438, whose nucleus has been classified as a low-ionization nuclear emission-line region (LINER). The data are supplemented by mid-infrared imaging, and compared to previous HST broad-band images. Images and position-velocity maps of the [FeII] and H₂ line emissions are presented and compared with the distribution of the optical narrow-line region and radio features. Our results show that shocks (possibly driven by a radio jet) contribute to an important fraction of the excitation of [FeII], while X-ray heating from a central active galactic nucleus (AGN) may be responsible for the H₂ excitation. We address the question whether the outflow has an AGN or a starburst origin by providing new estimates of the central star formation rate and the kinetic energy associated with the gas. By fitting a Sérsic bulge, an exponential disc and a compact nuclear source to the light distribution, we decomposed NGC4438's light distribution and found an unresolved nuclear source at 0.8arcsec resolution with M_K = -18.7 and J - H = 0.69. Our measured bulge velocity dispersion, 142 kms^-1, together with the standard relation, suggests a central black hole mass of . The stellar kinematics measured from the NIR CO lines show a strong peak in the velocity dispersion of sigma₀ ~ 178 kms^-1 in the central 0.5arcsec, which is possible kinematic evidence of a central black hole. We calculated a general expression for the integrated Sérsic profile flux density in elliptical geometry, including the case of `discy' isophotes.