  We compute the statistical entropy of a BTZ black hole in the context of
three-dimensional Euclidean loop quantum gravity with a cosmological constant
$\Lambda$. As in the four-dimensional case, a quantum state of the black hole
is characterized by a spin network state. Now however, the underlying colored
graph $\Gamma$ lives in a two-dimensional spacelike surface $\Sigma$, and some
of its links cross the black hole horizon, which is viewed as a circular
boundary of $\Sigma$. Each link $\ell$ crossing the horizon is colored by a
spin $j_\ell$ (at the kinematical level), and the length $L$ of the horizon is
given by the sum $L=\sum_\ell L_\ell$ of the fundamental length contributions
$L_\ell$ carried by the spins $j_\ell$ of the links $\ell$. We propose an
estimation for the number $N^\text{BTZ}_\Gamma(L,\Lambda)$ of the Euclidean BTZ
black hole microstates (defined on a fixed graph $\Gamma$) based on an analytic
continuation from the case $\Lambda>0$ to the case $\Lambda<0$. In our model,
we show that $N^\text{BTZ}_\Gamma(L,\Lambda)$ reproduces the Bekenstein-Hawking
entropy in the classical limit. This asymptotic behavior is independent of the
choice of the graph $\Gamma$ provided that the condition $L=\sum_\ell L_\ell$
is satisfied, as it should be in three-dimensional quantum gravity.
