A quick look at Ed Spence's page reveals two such examples: (7,3,3) and (16,6,3).

If there is a known classification and/or name by which such designs go, I'd love to know about them too.

EDIT: I am specifically interested in designs where at least one pair of blocks has a nonempty intersection, with multiple blocks allowed. A slightly off-the-beaten-path kind of object, perhaps.

EDIT2: Oops, I actually meant empty intersection. (Nevertheless, the case $B=V$ is too trivial for my purposes.) I am afraid that this creates a problem with Yuichiro Fujiwara's beautiful construction. Really sorry about the mistake, I've only noticed it now.