Does ICAO allow continued flight in RVSM airspace after a transponder failure?
Note: I assume bare ICAO requirements in this answer. Most states that have RVSM implementations, as well as operator approvals for RVSM, will have additional stricter rules that will change the answer for specific cases. 



There are no requirements in the ICAO Annexes and SARPS that require and aircraft the leave the RVSM airspace if the transponder fails. 

Note that RVSM airspace is not necessarily airspace where ATS Surveillance services are offered, so for ATC purposes, a functioning transponder is not strictly necessary in RVSM airspace. 

The only other use of the transponder that could prevent flying in RVSM airspace (under bare ICAO regulations) is the Airborne Collision Avoidance System (ACAS/TCAS). There are no requirements to have a functioning ACAS/TCAS in ICAO RVSM requirements, so no transponder is needed in support of such requirement. 



Even if a transponder would be necessary to enter RVSM airspace (which is the case in for example EASA regulations), a transponder failure while being in RVSM airspace does (under ICAO rules) does not necessarily mean that the flight has to leave that airspace ASAP. 

The following is from ICAO Doc 4444 (PANS ATM). 


  8.8.3.3    AIRCRAFT TRANSPONDER FAILURE IN AREAS WHERE THE CARRIAGE OF A FUNCTIONING TRANSPONDER IS MANDATORY 
  
  8.8.3.3.1    When an aircraft experiencing transponder failure after departure is operating or expected to operate in an area where the
  carriage of a functioning transponder with specified capabilities is
  mandatory, the ATC units concerned should endeavour to provide for
  continuation of the flight to the aerodrome of first intended landing
  in accordance with the flight plan. However, in certain traffic
  situations, either in terminal areas or en-route, continuation of the
  flight may not be possible, particularly when failure is detected
  shortly after take-off. The aircraft may then be required to return to
  the departure aerodrome or to land at the nearest suitable aerodrome
  acceptable to the operator concerned and to ATC. 
  
  ...




Flight in Class A airspace without transponder is possible under ICAO rules, as long as the airspace is not surveillance airspace. Procedural air traffic control does not require surveillance, and a transponder is therefor not required.