Clojure multiline regular expression
You need to use the (?s) "dotall mode" switch.
Example:
user=> (re-find #"\d{3}.\d{3}" "123\n456")
nil
user=> (re-find #"(?s)\d{3}.\d{3}" "123\n456")
"123\n456"
The (?m) switch is deceptively named -- it changes what the ^ and $ anchors do, allowing them to also match start-of-line and end-of-line, respectively -- which is not want you want.
Disclaimer: I'm not a Clojure programmer, but I think this problem is independent of the language.
When multi-line mode is enabled, the interpretation of the caret ^ and the dollar $ changes like this: Instead of matching the beginning and end of the entire input string, they match the beginning and the end of each line in the input string. This is - as far as I can see - not what you want/need.
What you want is for your .*s to match newlines (what they don't do by default) and this can be done by enabling the single-line mode (aka dot-all mode). So this means:
(re-find #"(?is)^<html>.*<body>.*</body>.*</html>" c))
You can also verify this on RegExr.