  We perform the first comprehensive constraints on two Finslerian models,
i.e., the simplest Finslerian $\Lambda$CDM (F$\Lambda$) and non-flat F$\Lambda$
models using the Type Ia supernovae, baryonic acoustic oscillations, cosmic
microwave background and lensing observations. Using the most stringent
constraints we can provide, we find that the constrained typical parameters of
both Finslerian models are all consistent with zero at the $2\sigma$ confidence
level (C.L.). This means that both Finslerian models just deviate very slightly
from the $\Lambda$CDM one, which is also verified by using two geometrical
diagnostics to distinguish three different models from each other. We also find
that a spatially flat universe is still preferred in the framework of Finsler
geometry, that our measured values of the spectral index $n_s$ of primordial
power spectrum in both Fisnlerian models exclude the scale invariance at more
than $8\sigma$ C.L., and that the current $H_0$ tension can be relieved from
$3.4\sigma$ to $2.9\sigma$ and $2.8\sigma$ in the F$\Lambda$ and non-flat
F$\Lambda$ models, respectively.
