  We consider the proof system Res($\oplus$) introduced by Itsykson and Sokolov
(Ann. Pure Appl. Log.'20), which is an extension of the resolution proof system
and operates with disjunctions of linear equations over $\mathbb{F}_2$.
  We study characterizations of tree-like size and space of Res($\oplus$)
refutations using combinatorial games. Namely, we introduce a class of
extensible formulas and prove tree-like size lower bounds on it using
Prover-Delayer games, as well as space lower bounds. This class is of
particular interest since it contains many classical combinatorial principles,
including the pigeonhole, ordering, and dense linear ordering principles.
  Furthermore, we present the width-space relation for Res($\oplus$)
generalizing the results by Atserias and Dalmau (J. Comput. Syst. Sci.'08) and
their variant of Spoiler-Duplicator games.
