I think this holds in quite some generality by the following simple argument. Let $S$ be a Polish space, let's say. If $T(t)$ is a Markov-Feller semigroup on $C_b(S)$ with kernel $p_t(x,dy)$, then note that for $f \in C_b(S) \cap L^1(\mu)$,
\begin{equation}
||T(t)f||_{L^p(\mu)}^p = \int_S \left| \int_S p_t(x, d y) f(y) \right|^p \mu(dx) \leq \int_S \left(\int_S p_t(x,dy) |f(y)|^p\right) \mu(dx) = ||f||_{L^p(\mu)}^p,
\end{equation}
which shows that $(T(t))$ is contracting with respect to the $L^p(\mu)$-norm.
Approximation of $f \in L^p(\mu)$ by continuous functions should then give you the extension you are looking for.

Note, $L^p(\mu)$ for $p < 1$ is not a Banach space so talking about strongly continuous semigroups does not seem to make sense.

Update: simplified argument.