Answer ... Yes, they can be appealed, but the filing of an appeal does not stay execution of the foreign judgment as a Cyprus court judgment.
If the defendant wishes to suspend execution, it can apply for a stay of execution pending appeal, but the court will grant such stay of execution subject to conditions (eg, the lodging with the registrar of the court of a bank guarantee for all sums payable under the foreign judgment to secure the same until the appeal is heard).
Answer ... Under Cypriot law, there is no right for an injunction pending an appeal or for obtaining a so-called Erinford injunction.