  We consider notions of space complexity for Infinite Time Turing Machines
(ITTMs) that were introduced by B. L\"owe and studied further by J. Winter.
  We answer several open questions about these notions, among them whether low
space complexity implies low time complexity (it does not) and whether one of
the equalities P=PSPACE, P$_{+}=$PSPACE$_{+}$ and P$_{++}=$PSPACE$_{++}$ holds
for ITTMs (all three are false). We also show various separation results
between space complexity classes for ITTMs. This considerably expands our
earlier observations on the topic in section 7.2.2 of \cite{Ca2}, which appear
here as Lemma $6$ up to Corollary $9$.
