I recall seeing a show years ago. fIREHOSE was doing a "50 states in 50 days" tour, and this show was a triple bill with Madder Rose opening. Madder Rose's bassist popped his E string, and he just knocked the bassline up an octave, which really kicked the song into fifth gear. He then spent minutes trying to beg, borrow or steal a bass or bass string, which just knocked all the momentum out of the show.
Later, Mike Watt popped his G string. (I still have it.) He nodded to George Hurley, who started in on a drum solo. Maybe a drum solo isn't the most punk-rock thing ever, but it filled the space while Mike changed a string and tuned up. At the end of it, there's Mike Watt, Greg Norton (ex-Husker Du bassist) and me, watching George get his thing on, with Mike looking at me and nodding toward George, as if to say, "That's my drummer. Isn't he good?" And he was.
The point of it is, it happens. It happens to starting out opening bands and it happens to top-of-bill bands. Things fall apart. It's scientific. But there are a few takeaway points.
1) Be able to do without - The first bassist really took the song to the next level just by knocking the bass up an octave. Word is that Jeff Beck dropped a pick, started playing with fingers and decided to always go that way. There might be things you can't do by picking with fingernails, but maybe what you can do will work out for a while.
2) Have spares - the SEALs say "Two is one. One is none." Have spare strings. Have spare picks. Have spare cables. If you need it, have two in case one breaks. If you play a Floyd Rose, a getting the balance back will take longer than a song, so bring a second guitar.
That being said, it's a good idea to have several spare windings on the tuner of Floyd Rose guitars, so if the break is near the bridge, you can unlock the bridge and the nut, run some more string, and get it back together fairly quickly.
3) Have a plan - In my story, Mike Watt went to the drum solo when he had to change strings. On the Dead Kennedys' Give Me Convenience or Give Me Death, there's a live track called "Night of the Living Rednecks" where, after East Bay Ray had a string break or something, the rhythm section went into a jazz groove while Jello Biafra told a story about being a punk being harrassed by rednecks, and it's my favorite track on that album. Anything is better than 5 minutes of silence while you try to get your act together.
4) Take it up a notch - 10s are a little sturdier than 9s, which is why I tend to go 10s.
5) Take it down a notch - Do you need to strum that hard? It may look like you're playing all intense, but there's only so hard you can pluck a string before it makes no difference. There's other sorts of stagecraft to look all hard and intense and be easier on your strings.
6) Don't play with knives - I used to have these Hot Licks brass picks from Dunlop. They're thin, and they sound bright and jangly, but they're metal and don't bend, unlike plastic thin picks. I was playing acoustic rhythm in an otherwise electric rock band, with in-ear monitors, and I simply could not hear myself unless I developed a strong picking technique, which I've been trying to dial back since. But I was carving the windings off my wound G string, because I was in effect sharpening this pick to get really sharp. I like it, and I use it on electric where I have a much more mellow strumming style because I have amplification backing me up, but I moved to an unwound G on an acoustic just so I wouldn't have to change strings so often, just because I didn't realize my pick was carving up my strings.
Other players may have other issues. Burrs in the saddle are common reasons why strings break. If you have a recurring string-breaking problem, the problem might be with your gear.
7) It is important - I saw Madder Rose two years later. They had another bassist. I don't know the story, but clearly, if you can play but you let the band down enough when they need you, they can find another person who can play.