Context. Deep spectroscopic surveys with the Atacama Large Millimeter/submillimeter Array (ALMA) have revealed that some of the brightest infrared sources in the sky correspond to concentrations of submillimeter galaxies (SMGs) at high redshift. Among these, the SPT2349-56 protocluster system is amongst the most extreme examples given its high source density and integrated star formation rate. Aims. We conducted a deep Lyman-alpha line emission survey around SPT2349-56 using the Multi-Unit Spectroscopic Explorer (MUSE) at the Very Large Telescope (VLT) in order to characterize this uniquely dense environment. Methods. Taking advantage of the deep three-dimensional nature of this survey, we performed a sensitive search for Lyman-alpha emitters (LAEs) toward the core and northern extension of the protocluster, which correspond to the brightest infrared regions in this field. Using a smoothed narrowband image extracted from the MUSE datacube around the protocluster redshift, we searched for possible extended structures. Results. We identify only three LAEs at z = 4.3 in this field. This number is lower than expectations for blank fields, and is significantly lower than the number of previously identified SMGs in this system. We find an extended Lyman-alpha structure of about 60 x 60 kpc(2) in size located 56 kpc east of the protocluster core. Three SMGs coincide spatially with the location of this structure. All the other SMGs are undetected in Lyman-alpha emission, which is consistent with the conspicuous dust obscuration in these systems. We conclude that either the three co-spatial SMGs or the protocluster core itself are feeding ionizing photons to the Lyman-alpha structure.