  We use the kinematic data of the stars in eight dwarf spheroidal galaxies to
assess whether $f(R)$ gravity can fit the observed profiles of the
line-of-sight velocity dispersion of these systems without resorting to dark
matter. Our model assumes that each galaxy is spherically symmetric and has a
constant velocity anisotropy parameter $\beta$ and constant mass-to-light ratio
consistent with stellar population synthesis models. We solve the spherical
Jeans equation that includes the Yukawa-like gravitational potential appearing
in the weak field limit of $f(R)$ gravity, and a Plummer density profile for
the stellar distribution. The $f(R)$ velocity dispersion profiles depend on two
parameters: the scale length $\xi^{-1}$, below which the Yukawa term is
negligible, and the boost of the gravitational field $\delta>-1$. $\delta$ and
$\xi$ are not universal parameters, but their variation within the same class
of objects is expected to be limited. The $f(R)$ velocity dispersion profiles
fit the data with a value $\xi^{-1}= 1.2^{+18.6}_{-0.9}$ Mpc for the entire
galaxy sample. On the contrary, the values of $\delta$ show a bimodal
distribution that picks at $\bar{\delta}=-0.986\pm0.002$ and
$\bar{\delta}=-0.92\pm0.01$. These two values disagree at $6\sigma$ and suggest
a severe tension for $f(R)$ gravity. It remains to be seen whether an improved
model of the dwarf galaxies or additional constraints provided by the proper
motions of stars measured by future astrometric space missions can return
consistent $\delta$'s for the entire sample and remove this tension.
