  The aim of this note is to show, by providing an elementary way to construct
counter-examples, that the strong Feller and the joint (space-time) continuity
for a semigroup of Markov kernels on a Polish space are not enough to ensure
the existence of an associated c\`adl\`ag Markov process on the same space. One
such simple counter-example is the Brownian semigroup on $\mathbb{R}$
restricted to $\mathbb{R}\setminus \{0\}$, for which it is shown that there is
no associated c\`adl\`ag Markov process. Using the same idea and results from
potential theory we then prove that the analogous result with c\`adl\`ag Markov
process replaced by right Markov process also holds, even if one allows to
change the Polish topology to another Polish topology with the same Borel
$\sigma$-algebra.
