Try unplugging your Sonos speakers from power for a couple of minutes and reboot your router and phone.

Trouble is, it’s four different speakers (Beam, Play 1, etc.). They all drop the connection simultaneously. And all my other wireless devices (phone, laptop) aren’t dropping their wifi.

Sounds a lot like, at least potentially, a duplicate IP problem. Since the Sonos speakers all communicate with each other routinely, if the particular issue is with the one speaker that you’re “starting” playback on, and the others are grouped with it, it could certainly cause that sort of behaviour.  I’d start, at the very least, by refreshing your Sonos connections in your router by unplugging all the Sonos devices from power, and while they’re unplugged, rebooting your router. Once the router comes back up, go ahead and power up the Sonos devices, starting with the one (if there is one) connected to the router with an ethernet cable. If there isn’t, then any order is fine. 

Test after they’ve all booted up, and see if you can recreate the “drop” in connection. If so, then submit a system diagnostic within 10 minutes of experiencing this problem, and call Sonos Support to discuss it.

There may be information included in the diagnostic that will help Sonos pinpoint the issue and help you find a solution.

When you speak directly to the phone folks, they have tools at their disposal that will allow them to give you advice specific to your Sonos system and network.