Consider the stationary (i.e, independent of time) Stokes equations $$\mathrm{div}~ \sigma = f$$ where $\sigma$ is the stress tensor, $f$ is the external force.
Denote by $M,L,T$ the mass, length, and time, respectively. Then the dimension of $\sigma$ is $ML^{-1}T^{-2}$, which leads to the dimension of $\mathrm{div}~\sigma$ is $ML^{-2}T^{-2}$. However, it contradicts to the dimension of force $f$, which is $MLT^{-2}$.
Can anyone explain me the discrepancy?
The units of $\text{div}(\sigma)$ (taken with respect to cartesian components) is $ML^{-2}T^{-2}$.
The units of force $F$ is $MLT^{-2}$, as you said. But note that the units of (volume) force density $f$ is $MLT^{-2} / L^3 = ML^{-2}T^{-2}$
The misunderstanding you're having is that in the equation $\text{div}(\sigma) = f$, the $f$ is not force, but rather the (volume) force density, which is consistent with the dimensional analysis above.
By the way, the same is true also in electrodynamics, where you have a similar equation $\text{div}(\sigma) = f$, (assuming the Poynting vector $S$ doesn't depend on time; i.e the "static" case), where $\sigma$ is the Maxwell stress tensor, and $f = \rho E + J \times B$ is the (volume) electromagnetic force density.
Again, both sides of this equation have units of $ML^{-2}T^{-2}$.