  Let $p$ be a prime number, $F $ a non-archimedean local field with residue
characteristic $p$, and $R$ an algebraically closed field of characteristic
different from $ p$. We thoroughly investigate the irreducible smooth
$R$-representations of $SL_2(F)$. The components of an irreducible smooth
$R$-representation $\Pi$ of $GL_2(F)$ restricted to $SL_2(F)$ form an
$L$-packet $L(\Pi)$. We use the classification of such $\Pi$ to determine the
cardinality of $L(\Pi)$, which is $1,2$ or $4$. When $p=2$ we have to use the
Langlands correspondence for $GL_2(F)$. When $\ell$ is a prime number distinct
from $p$ and $R=\mathbb Q_\ell^{ac}$, we establish the behaviour of an integral
$L$-packet under reduction modulo $\ell$. We prove a Langlands correspondence
for $SL_2(F)$, and even an enhanced one when the characteristic of $R$ is not
$2$. Finally, pursuing a theme of \cite{HV23}, which studied the case of inner
forms of $GL_n(F)$, we show that near identity an irreducible smooth
R-representation of $SL_2(F)$ is, up to a finite dimensional representation,
isomorphic to a sum of $1,2$ or $4$ representations in an $L$-packet of size
$4$ (when $p$ is odd there is only one such $L$-packet).
