We investigate the following question: does there exist a compatible extension of a given compatible partial metric $p:A^2\to\mathbb R$ on a closed subset $A$ of a partially metrizable space $X$? We obtain a positive answer to this question in the case when the corresponding quasi-metric $q_p(x,y)=p(x,y)-p(x,x)$ has an extension that generates a weaker topology on $X$ (in particular, if $q_p$ is bounded). Moreover, we give an example which shows that in general the answer to the question is negative.