while lava is liquid, it still has the density of rock, so any living thing being heavy enough displace it to allow any noticeable amount of sinkage is extremely unlikely.

1. The density of lava depends largely on the amount of gases in it. Some lavas look more like viscous foam than molten rock.
2. The possibility of immersion depends not only on density, but also on dynamic characteristics of meduim. Thus, a person can drown in quicksand, although the density of the sand is very high. You can also take mud baths, although the density of therapeutic mud is comparable to the density of the lightest carbonatite lavas.