Tensor networks are increasingly known as valuable tools for bypassing the curse of dimensionality present while solving a wide range of differential equations. The simplest kind of tensor networks are the Matrix Product States (MPS), or Quantized Tensor Trains, which factorize large vectors as a linear chain of smaller tensors. In a compatible manner, Matrix Product Operators (MPO) factorize linear operators, that can be efficiently applied to MPS within the tensor-network representation. Recently, we have released the version 3.0 of the SeeMPS Python library~[1], aimed at providing a simple framework to operate with MPS and MPO as if they were standard vector and matrix arrays. This library also offers a LAPACK-like suite of tools that enable higher level computations ranging from eigenvalue search and fourier transforms, to the solution of linear systems. Moreover, SeeMPS presents a selection of advanced numerical analysis tools for the integration and differentiation of functions encoded in MPS, providing means to efficiently discretize and solve linear and nonlinear differential equations, whenever the correlations in the system allow it. As a proof of principle, we demonstrate some capabilities of SeeMPS by resolving long-ranged, collisionless plasma dynamics represented by the Vlasov-Poisson system of equations. Similarly to previous tensor network studies~[2], we split the Vlasov equation into successive linear and nonlinear advections which are solved with TDVP~[3]. The discretization of derivatives, as well as the solution of the coupled Poisson equation, is accomplished with pseudospectral accuracy, directly in real space, using Hermite Distributed Approximating Functionals [4, 5].