In a press release, the USPTO stated it has sufficient available funds not tied to the current fiscal year that it will be able to remain open for six days following any shutdown of government functions. Based on the current appropriations, that would put the USPTO open until April 18, assuming a shutdown begins April 11. After that time, limited functions should still be available, including accepting new electronically-filed applications and other IT functions. It is not clear what, if any, other services will continue to be available after that time.
To read the full press release, click here.