We extend the recently developed Jacobi no-core shell model to hypernuclei. Based on the coefficients of fractional parentage for ordinary nuclei, we define a basis where the hyperon is the spectator particle. We then formulate transition coefficients to states that single out a hyperon-nucleon pair which allow us to implement a hypernuclear many-baryon Hamiltonian for $p$-shell hypernuclei. As a first application, we use the basis states and the transition coefficients to calculate the ground states of $^{4}_{Lambda}$He, $^{4}_{Lambda}$H, $^{5}_{Lambda}$He, $^{6}_{Lambda}$He, $^{6}_{Lambda}$Li, and $^{7}_{Lambda}$Li and, additionally, the first excited states of $^{4}_{Lambda}$He, $^{4}_{Lambda}$H, and $^{7}_{Lambda}$Li. In order to obtain converged results, we employ the similarity renormalization group (SRG) to soften the nucleon-nucleon and hyperon-nucleon interactions. Although the dependence on this evolution of the Hamiltonian is significant, we show that a strong correlation of the results can be used to identify preferred SRG parameters. This allows for meaningful predictions of hypernuclear binding and excitation energies. The transition coefficients will be made publicly available as HDF5 data files.
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