Unfortunately no! It is referred to as Marker Bounce Effect. This is in the wish-list for Leaflet but not implemented yet.
If you up-vote it there, perhaps we can see this feature soon, or if you are really up for it, you could build your own bouncy marker as a 3rd party plugin. Such plugins can be created by extending the core Leaflet.js. Refer to one such example MarkerCluster in github.