  We construct two new classes of analytical solutions in three-dimensional
spacetime and in the framework of $f(R)$ gravity. The first class represents a
non-rotating black hole (BH) while the second class corresponds to a rotating
BH solution. The Ricci scalar of these BH solutions have non-trivial values and
are described by the gravitational mass $M$, two angular momentums $J$ and
$J_1$, and an effective cosmological constant $\Lambda_{eff}$. Moreover, these
solutions do not restore the $3$-dimensional Ba\~{n}ados-Teitelboim-Zanelli
(BTZ) solutions of general relativity (GR) which implies the novelty of the
obtained BHs in $f(R)$ gravity. Depending on the range of the parameters, these
solutions admit rotating/non-rotating asymptotically AdS/dS BH interpretation
in spite that the field equation of $f(R)$ has no cosmological constant.
Interestingly enough, we observe that in contrast to BTZ solution which has
only causal singularity and scalar invariants are constant everywhere, the
scalar invariants of these solutions indicate strong singularity for the
spacetime. Furthermore, we construct the forms of the $f(R)$ function showing
that they behave as polynomial functions. Finally, we show that the obtained
solutions are stable from the viewpoint that heat capacity has a positive
value, and also from the condition of Ostrogradski which state that the second
derivative of $f(R)$ should have a positive value.
