  The processes operating in blazar jets are still an open question. Modeling
the radiation emanating from an extended part of the jet allows one to capture
these processes on all scales. Kinetic codes solving the Fokker-Planck equation
along the jet flow are well suited to this task, as they can efficiently derive
the radiation and particle spectra without the need for computationally
demanding plasma-physical simulations. Here, we present a new extended
hadro-leptonic jet code -- ExHaLe-jet -- which considers simultaneously the
processes of relativistic protons and electrons. Within a pre-set geometry and
bulk flow, the particle evolution is derived self-consistently. Highly
relativistic secondary electrons (and positrons) are created through
$\gamma$-$\gamma$ pair production, Bethe-Heitler pair production, and pion/muon
decay. These secondaries are entrained in the jet flow decreasing the ratio of
protons to electrons with distance from the jet base. For particle-photon
interactions, we consider all internal and many external photon fields, such as
the accretion disk, broad-line region, and the dusty torus. The external fields
turn out to be the most important source for particle-photon interactions
governing the resulting photon and neutrino spectra. In this paper, we present
the code and an initial parameter study, while in follow-up works we present
extensions of the code and more specific applications.
