You're asking about L2 broadcast domains, not L3 broadcast domains, correct?
As the other posters have noted, if the L2 switch supports VLANs, it's possible a L2 switch has multiple L2 broadcast domain because each VLAN has a broadcast domain.
But what might make this easier to understand is if we go back to Ethernet before there were switches, all L2 frames transversed the same "wire" (see 10Base2 and 10Base5).
Having all frames on the same "wire" can fill the capacity of a wire, even from just two hosts, often fairly easily. To provide additional capacity, without getting into L3 (very slow and expensive in ye olden times), something called a "bridge" came into existence. This supported two "wires", transmitting frames from one "wire" to another, but not if the bridge "knew" the source and destination were already on the same "wire". L2 broadcast, though, have a destination to all devices on the same "wire", which also included "wires" connected by bridges. So, all hosts logically reachable by a L2 broadcast are in the same L2 domain.
A L2 switch is actually also known as a multi-port bridge. It keeps track of all host MACs, seen as source MACs, on all its connected ports. If a host is sending to a MAC on the same port it does not forward it (like the earlier bridges). If a host is sending to a MAC known to be on a different port, it transmits the frame on just that port. If the switch doesn't know where the destination is for the destination MAC it's a L2 broadcast, it transmits to all the ports except the port that received the frame. For L2 broadcasts, again, they are sent and seen on the whole L2 broadcast domain.
If a L2 switch does not support VLANs, there will only be one L2 broadcast domain, but other switches can connect to each other, just like it, so the same broadcast domain, again even for not VLAN switches, can include multiple switches.
