Updating the response, since Firefox now supports Webp.
Support
Firefox has image support .webp
since January 2019, from version 65 (see the ad). It is possible to see all browsers that support .webp
on the website Can I use.... In the current versions listed, only Internet Explorer, Safari, iOS Safari and Kaios Browser do not have this support.
Support status in June/2020:
1 Partial support. Refers to not supporting lossless images (Lossless), alpha and animated Webp images.
2 Partial support. Refers to no support of animated Webp images.
About the Webp
About Webp being experimental, the answer is no. It was created on September 30, 2010, being experimental at the time, but in 2018 (date of the question) it was already something "concrete". I couldn’t find a date when it came to being considered "in production", but the first version of Google Chrome to support it (according to Can I use) was 32, released in 2014.
Webp is just a format like all the others, but it has features that made it increasingly adopted on the Internet, to be able to transfer images with fewer bytes with similar quality, more information can be found on Google website, in the FAQ and in this article. It is worth noting that Webp is an Open Source format.
And Internet Explorer and Safari?
Good, since the Internet Explorer no longer has Microsoft support, we can see that the Edge supports Webp, then Microsoft did "her part".
About Apple, despite some news from 2016 (cnet and Macmagazine) report that Apple was testing Webp support, nothing new has emerged since.
Some discussions were generated in the Apple forum questioning why not have this support (1, 2, 3 and 4), but no concrete answer. Just like it was with Firefox, only time will tell if Safari will have it, but probably not yet implemented because this format was created by Google and companies have their "intrigues".
It seems that firefox does not have webp support yet. But it is possible to make it work: http://www.habilelabs.io/webp-format-speed-improvement-making-cross-broswer-compatible/
– user28595
Apparently there are few that have support, take a look at the caniuse https://caniuse.com/#search=webp
– Jackson
https://www.reddit.com/r/firefox/comments/46wxye/why_is_firefox_still_not_supporting_webp/
– hugocsl