The VLT/SPHERE instrument includes a unique long-slit spectroscopy (LSS) mode coupled with Lyot coronagraphy dedicated to the spectral characterization of directly imaged giant exoplanets. The performance of this mode is limited by its non-optimal coronagraph, but in a previous work we demonstrated that it could be significantly improved at small inner-working angles using the stop-less Lyot coronagraph (SLLC). A prototype of the SLLC was installed in VLT/SPHERE in 2014 during the reintegration of the instrument in Paranal, and it was extensively tested in 2015 to characterize its performance. The performance is tested in both imaging and spectroscopy using data acquired on the internal source of SPHERE. In imaging, we obtain a raw contrast gain of a factor 10 at 0.3" with the SLLC. We also demonstrate that no Lyot stop is required to reach the full performance, which validates the SLLC concept. Comparison with a realistic simulation model shows that we are currently limited by the internal phase aberrations of SPHERE. In spectroscopy, we obtain a gain of 1 mag in a limited range of angular separations. Simulations show that although the main limitation comes from phase errors, the performance in the non-SLLC case is very close to the ultimate limit of the LSS mode. We present the very first on-sky data with the SLLC, which appear extremely promising for the future scientific exploitation of an apodized LSS mode in SPHERE. Finally, we explore a new possibility for the speckle subtraction in the LSS mode that could significantly improve the data analysis with respect to methods based on spectral differences.