Did Pudgalavada Buddhists claim the the "person" existed between lives and / or in one life? Did Pudgalavada Buddhists claim the the "person" existed between lives and / or in one life?

The Pudgalavadins were an old school of Buddhists who thought there was a person. I'm currently unclear whether that is ultimately, in the sense that Theravada and Mahayana Buddhists use the term 'ultimate'.

They debated about this with e.g. the Vibhajyavada sect, which I think later became Theravada Buddhism. You can read about the canonical dismissal of it in Theravada Buddhism in the 1st chapter of their Kathavatthu.



To clarify, one of my links say 


  It is this self, they maintained, that dies and is reborn through
  successive lives in Samsara, continuing to exist until enlightenment
  is attained.


Is this the same person, each life?
I would not go as far as to call Pudgalavada "a school", as with most early Buddhism there were people who held certain position, that later received a name and was used for comparison with mainstream traditions.

If I remember correctly, Pudgalavadins claimed that person exists for all practical intents and purposes, so I think that means conventional existence, not ultimate. You should be able to find more details on Pudgalavada in Ch. 9 of Abhidharmakosa-Bhasya of Vasubandhu. See this for a quick summary - pp. 89-91:


  The Pudgalavadins introduce the notion that persons are conceived on the basis of the skandhas as fire is conceived in reliance upon fuel without being other than or the same in existence as fuel. 
  Pudgalavadins assert that in addition to phenomena that are conditioned &amp; unconditioned (samskrta/asamskrta),
  impermanent and permanent (anitya/nitya), the 12 ayatanas, the 18 dhatus, etc., there are, in addition, phenomena that are
  inexplicable (avakrtavya). For the Pudgalavadins, conventional realities may be substantially established (as they are for the
  Vaibhasikas) or inexplicable. Inexplicable entities are entities without separate identities – they are neither the same as, nor
  different from, the skandhas. The Pudgalavadins reject the logic that everything is either substantially real or substantially
  established. Inexplicable persons are conventional realities insofar as they are conceived in dependence upon the skandhas,
  but ultimately exist insofar as they exist apart from being conceived. Persons are single entities without separate identities, an
  inexplicable unity.
  The Pudgalavadins maintain that the person is perceived through an inexplicable
  perception incidental to a consciousness perceiving its proper object. It is an inexplicable perception because it is neither the
  same nor different from the perception of the object.
  The basic attitude of Vasubandhu, and others, toward
  the Pudgalavadins is that in their theory of persons, it can appear that the person is substantially real, despite attempts by the
  Pudgalavadins to refute this attribution.


Mahayana, in particular Tibetan Nyingma tradition, insists that to say that person conventionally exists still leaves the mind grasping after the person, so more for didactic than for analytical purposes they say that there is no person is the right view. See "Mipham's Beacon of Certainty" (Mipham uses the traditional rhetoric of "snake", "vase" etc. but what he means is person, or entity in the broad sense)

Also see Satyasiddhisastra of Harivarman pages 67-74 for various other views around Pudgala and their refutations. As of late I consider Satyasiddhisastra a key text providing the missing link between Early Buddhism and Mahayana, with Abhidharmakosa and Mulamadhyamakakarika correspondingly on one and the other sides of the divide (conceptually, not chronologically).