  In a (2+1)-dimensional spacetime with a negative cosmological constant, the
thermodynamics and the entropy of an extremal rotating thin shell, i.e., an
extremal rotating ring, are investigated. The outer and inner regions are taken
to be the Ba\~{n}ados-Teitelbom-Zanelli (BTZ) spacetime and the vacuum ground
state anti-de Sitter (AdS) spacetime, respectively. By applying the first law
of thermodynamics to the extremal shell one shows that its entropy is an
arbitrary function of the gravitational area $A_+$ alone, $S=S(A_+)$. When the
shell approaches its own gravitational radius $r_+$ and turns into an extremal
rotating BTZ black hole, it is found that the entropy of the spacetime remains
such a function of $A_+$. It is thus vindicated, that extremal black holes,
here extremal BTZ black holes, have different properties from the corresponding
nonextremal black holes, which have the Bekenstein-Hawking entropy $S(A_+)=
\frac{A_+}{4G}$, where $G$ is the gravitational constant. It is argued that for
the extremal case $0\leq S(A_+)\leq \frac{A_+}{4G}$. Thus, rather than having
just two entropies for extremal black holes, as previous results debated, 0 and
$\frac{A_+}{4G}$, it is shown that extremal black holes may have a continuous
range of entropies, limited by precisely those two entropies. Surely, the
entropy that a particular extremal black hole picks must depend on past
processes, notably on how it was formed. It is also found a remarkable relation
between the third law of thermodynamics and the impossibility for a massive
body to reach the velocity of light. In the procedure, it becomes clear that
there are two distinct angular velocities for the shell, the mechanical and
thermodynamic angular velocities. In passing, we clarify, for a static
spacetime with a thermal shell, the meaning of the Tolman temperature formula
at a generic radius and at the shell. (Abridged version).
