  Using a thin shell, the first law of thermodynamics, and a unified approach,
we study the thermodymanics and find the entropy of a (2+1)-dimensional
extremal rotating Ba\~{n}ados-Teitelbom-Zanelli (BTZ) black hole. The shell in
(2+1) dimensions, i.e., a ring, is taken to be circularly symmetric and
rotating, with the inner region being a ground state of the anti-de Sitter
(AdS) spacetime and the outer region being the rotating BTZ spacetime. The
extremal BTZ rotating black hole can be obtained in three different ways
depending on the way the shell approaches its own gravitational or horizon
radius. These ways are explicitly worked out. The resulting three cases give
that the BTZ black hole entropy is either the Bekenstein-Hawking entropy,
$S=\frac{A_+}{4G}$, or it is an arbitrary function of $A_+$, $S=S(A_+)$, where
$A_+=2\pi r_+$ is the area, i.e., the perimeter, of the event horizon in (2+1)
dimensions. We speculate that the entropy of an extremal black hole should obey
$0\leq S(A_+)\leq\frac{A_+}{4G}$. We also show that the contributions from the
various thermodynamic quantities, namely, the mass, the circular velocity, and
the temperature, for the entropy in all three cases are distinct. This study
complements the previous studies in thin shell thermodynamics and entropy for
BTZ black holes. It also corroborates the results found for a (3+1)-dimensional
extremal electrically charged Reissner-Nordstr\"om black hole.
