In a lab setting it can be a while but out in the open it can't.
There also seems to be conflicting information about how long the virus survives in the air and on surfaces, some people say minutes and some information says it can survive for hours or days.
You would have to slice open your arm for a risk and that would be only if there was a large volume of blood that you were able to get into your huge cut.
If any of the infected fluid got into the irritated cuticle, or does a wound need to be gaping open and weeping because information seems to indicate open wounds, cuts, rashes etc like dermatitis. I suppose my concern was more the tap and the soap dispenser as I don't think anything from the sink actually splashed anywhere on my hands but I can't be sure. I feel really stupid that this has happened and also because I'm confused about transmission, the Internet has so much conflicting information.
Even if it enters your bloodstream?
Blood on a surface is not able to infect you with HIV.