Has any species of animal become extinct in the past 20 years?

Sadly, many. So many that scientists are talking about a massive extinction induced by humans, the sixth in the history of life:
Accelerated modern human-induced species losses: Entering the sixth mass extinction

The last animals added are:
2015 — Eastern Cougar, Puma Concolor Couguar
2013 – Formosan Clouded Leopard, Neofelis nebulosa brachyura
2012 — Pinta Island Tortoise, Chelonoidis abingdoni
2011 — Vietnamese Rhino, Rhinoceros sondaicus annamiticus

2009 — Christmas Island Pipistrelle, Pipistrellus murrayi
2007 — Chinese Paddlefish, Psephurus gladius
2007 — Yangtze River Dolphin, Lipotes vexillifer
etc, etc.
If you want to visit this sad list, here is the link:
Here's Every Single Animal That Became Extinct In The Last 100 Years
As a meta-analysis, a study published by experts from Stanford, Princeton and the University of  California-Berkeley declared the world’s vertebrates are going extinct  114 times faster than the natural rate of extinction.