Context. In most current debris disc models, the dynamical and the collisional evolutions are studied separately with N-body and statistical codes, respectively, because of stringent computational constraints. In particular, incorporating collisional effects (especially destructive collisions) into an N-body scheme has proven a very arduous task because of the exponential increase of particles it would imply. Aims. We present here LIDT-DD, the first code able to mix both approaches in a fully self-consistent way. Our aim is for it to be generic enough to be applied to any astrophysical case where we expect dynamics and collisions to be deeply interlocked with one another: planets in discs, violent massive breakups, destabilized planetesimal belts, bright exozodiacal discs, etc.Methods. The code takes its basic architecture from the LIDT3D algorithm for protoplanetary discs, but has been strongly modified and updated to handle the very constraining specificities of debris disc physics: high-velocity fragmenting collisions, radiation-pressure affected orbits, absence of gas that never relaxes initial conditions, etc. It has a 3D Lagrangian-Eulerian structure, where grains of a given size at a given location in a disc are grouped into super-particles or tracers whose orbits are evolved with an N-body code and whose mutual collisions are individually tracked and treated using a particle-in-a-box prescription designed to handle fragmenting impacts. To cope with the wide range of possible dynamics for same-sized particles at any given location in the disc, and in order not to lose important dynamical information, tracers are sorted and regrouped into dynamical families depending on their orbits. A complex reassignment routine that searches for redundant tracers in each family and reassignes them where they are needed, prevents the number of tracers from diverging.Results. The LIDT-DD code has been successfully tested on simplified cases for which robust results have been obtained in past studies: we retrieve the classical features of particle size distributions in unperturbed discs and the outer radial density profiles in ~r-1.5 outside narrow collisionally active rings as well as the depletion of small grains in dynamically cold discs. The potential of the new code is illustrated with the test case of the violent breakup of a massive planetesimal within a debris disc. Preliminary results show that we are able for the first time to quantify the timescale over which the signature of such massive break-ups can be detected. In addition to studying such violent transient events, the main potential future applications of the code are planet and disc interactions, and more generally, any configurations where dynamics and collisions are expected to be intricately connected.