I have a curious question. Is Identity-based encryption used in the wild?
In addition to the cases mentioned by Jack Lloyd and Bob Wall, the British government also seems to be deploying identity-based encryption, namely the MIKEY-SAKKE RFC standard. You can see for example this introduction, this page, and this FAQ on the government's website. Note that the key escrow feature of MIKEY-SAKKE (which is inherent to all IBEs) has been pictured by some as a backdoor (see e.g. here), which in my humble opinion is a bit unfair since the two notions are "close" but clearly distinct.
MIKEY-SAKKE seems to have been deployed at least in the context of emergency services, see this white paper. This makes sense, since this is a typical example where IBE can be helpful to simplify the PKI of an organisation and where the presence of a key escrow is not an issue.
SM9 (GM/T 0044.5‐2016) is an IBE scheme widely used in China. Unfortunately most of the documents are in Chinese with no English translations available. But a description of basic SM9 scheme is available on eprint and an open source implementation is in the OpenSSL fork GmSSL