Actually, the answer to your question, strictly addressing the exact way you asked, is YES. But, it requires clarification...
BBM is a BB-proprietary service, available only to BB users. As such, it requires subscription to, from your mobile service provider, an adequate data plan so as to enable the service. However, if you have the service enabled, it will "run" via WiFi...in other words, it does not require the carrier data network to be available in order to function. But, it does require subscription to an adequate data plan in order to function, regardless of pathway (e.g, carrier data network versus WiFi) used to operate it.
Occam's Razor nearly always applies when troubleshooting technology issues!
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