  Some extensions of the Standard Model of Particle Physics allow for Lorentz
invariance and Charge-Parity-Time (CPT)-invariance violations. In the neutrino
sector strong constraints have been set by neutrino-oscillation and
time-of-flight experiments. However, some Lorentz-invariance-violating
parameters are not accessible via these probes. In this work, we focus on the
parameters $(a_{\text{of}}^{(3)})_{00}$, $(a_{\text{of}}^{(3)})_{10}$ and
$(a_{\text{of}}^{(3)})_{11}$ which would manifest themselves in a non-isotropic
beta-decaying source as a sidereal oscillation and an overall shift of the
spectral endpoint. Based on the data of the first scientific run of the KATRIN
experiment, we set the first limit on $\left|(a_{\text{of}}^{(3)})_{11}\right|$
of $< 3.7\cdot10^{-6}$ GeV at 90\% confidence level. Moreover, we derive new
constraints on $(a_{\text{of}}^{(3)})_{00}$ and $(a_{\text{of}}^{(3)})_{10}$.
