Media playback is not supported on this device
Asher-Smith, 21, broke a bone in her foot in March and looked unlikely to make these championships as recently as a month ago.
But having run a season's best to get through the final, she knocked another half a second off her semi-final time as the Nethlerlands' Schippers held off Marie-Josee Ta Lou for gold with Shaunee Miller-Uibo coming through for bronze in London.
The great Brittney Reese won her fourth world long jump title in a thrilling final where the top four women were covered by only six centimetres.
There was disappointment for Britain's Lorraine Ugen, who had set the third biggest mark in qualifying but began with two fouls and had to settle for fifth with her fourth-round 6.72m.
But her compatriot Lynsey Sharp successfully appealed an initial disqualification to be dramatically reinstated into Saturday's 800m final.
Asher-Smith's finish was the fifth fourth place Great Britain have taken at these Worlds, Mo Farah's 10,000m gold on the first night the only medal they have won in the eight days so far.
Yet it would be hard to describe any of those individual results as a disappointment, and Asher-Smith was delighted to rescue something from a season that appeared wrecked.
"I was so happy to be here and just told myself to go out fast," she told BBC Sport.
"The crowd were getting louder and louder so I knew I was doing well, and I kept going for it.
"I'm frustrated, but quite happy to have run like that having broken my foot this year."
Schippers ran a season's best of 22.05 seconds as her strength down the home straight held off Ivorian Ta Lou, who also had to settle for silver in the 100m.
Miller-Uibo of Bahamas led the 400m until stumbling in the final few strides, but this was at least a little consolation for a gold that slipped away when it appeared hers.
Two-time world 110m hurdles champion Colin Jackson
Dina Asher-Smith is something incredible. The amount she has had to battle, both mentally and physically, is unbelievable.
22.22 seconds! I cannot describe to everybody how impressive that was.
She will be disappointed, but how much work she has missed over the summer with the broken foot...
She loves this event. It was an unbelievable performance through the rounds.
Media playback is not supported on this device
Reese's triumph earlier in the evening made her the first woman to win four world long jump titles.
After Darya Klishina - competing here as a neutral athlete after being cleared by the IAAF doping review board to circumvent the continuing ban on Russian athletes - had gone out to 6.78m in the first round, Serbia's Ivana Spanovic seized the lead with 6.96m with her second jump.
American Reese, the 2012 Olympic champion in this stadium five summers ago, then took control with her third-round 7.02m, before Klishina almost stole gold in the final round.
Her jump came up two centimetres short - although the silver was her first medal in a global final - and 2015 world champion Tianna Bartoletta snatched bronze with 6.97m on her final attempt.
Media playback is not supported on this device
Sharp looked to have made it through an 800m semi-final won by Olympic champion Caster Semenya as a fastest loser with one minute 59.47 seconds for fourth.
But she was disqualified when officials decided she had impeded US rival Charlene Lipsey.
Sharp's left arm had briefly hooked around the right arm of Lipsey in the final few strides, yet it appeared both inadvertent and to have given her no advantage.
The British team appealed against the decision and, an hour after leaving the track in tears, Sharp was reinstated.
Adelle Tracey could not join her despite running a personal best of 2:00.26 for sixth in her semi, while Shelayna Oskan-Clarke finished almost a second down on the time she ran in her first round heat as she too went out.
Media playback is not supported on this device
While Chris O'Hare qualified comfortably for the 1500m final by cruising through in fourth in his semi-final, Jake Wightman tired in the final 200m having been beautifully placed at the bell.
"Nobody is going into the final with any other expectation than trying to win a medal, so that's me too. That's the goal," said O'Hare.
Nick Miller briefly sat in silver position in the hammer final before being overtaken by his more storied rivals, his sixth place in 77.31m just 29cm off his personal best.
A thrilling 3,000m steeplechase final ended in an unlikely US one-two, Emma Coburn setting a new championship record of 9:02.58 and Courtney Frerichs smashing her personal best with 9:03.77 as defending champion Hyvin Jepkempoi was left trailing in third.
Fellow Kenyan Beatrice Chepkoech missed the water jump on the second lap and had to run back to clear it, then fell on the following lap to see her own chances tossed away.
Media playback is not supported on this device
Great Britain's Dina Asher-Smith came within seven hundredths of a second of a remarkable medal as Dafne Schippers retained her 200m title at the World Championships.