Can there be "one day becoming a fully-awakened Buddha" after Parinibbana?

Yes. Most Mahayana schools posit the one final vehicle (ekayana) and posit that everyone, in time, will become a fully enlightened buddha. As to the so-called Hinayana arhat, it is said that he has to enter the Mahayana path from the path of accumulation and go all the way to the path of no-more-learning, buddhahood. He has to go through the Mahayana path of accumulation, preparation, seeing, meditation.

According to Mahayana tenets, by way of generating bodhicitta, one enters the Mahayana path, whether one is established in Parinibbana (because the continuum is not severed), or abiding in nirvana with remainder, or an ordinary being.

Geshe Jamphel Gyaltsen's commentary on Shantideva's Bodhisattvacharyavatara says:

In comparison to someone who enters the Mahayana path directly, a Hinayana arhat faces more problems. In order to achieve self-liberation he has been meditating for a long time. He has gained a level of bliss and great happiness. Because of this habit of bliss and happiness it is very difficult for him to think about the suffering of sentient beings. It is not a spontaneous tendency for him to think about others´ suffering. So bodhicitta is not coming easily for him. Therefore, it is advisable to enter the Mahayana directly, even though it is not that easy, instead of going through the complete liberation of a Hinayana arhat and then enter the Mahayana. To illustrate this, there is a story about a practitioner who met Shariputra. Shariputra wanted to tell him about the achievement of a Hinayana arhat. Later on the practitioner met Manjushri who told him about the views of the Mahayana. Based on that he developed wrong view and as a result was reborn in the hells. What happened next is that, even though he was reborn in the hells, he still had the karmic imprint that connected him to the Mahayana path. Therefore it is said that even though he spent some time in the hells, it was still quicker for him to achieve the state of a buddha, than if he had gone through the stages shown by Shariputra. Therefore, Buddha Shakyamuni praised Manjushri for being so skilful in seeing that the person would react in a very heavy way, propelling him to the lower realms, yet that in this way he would achieve buddhahood quicker.

You might further study texts on 'Grounds and Paths' (Tib. Salam) as well as Asanga's bodhisattvabhumi.

I have no idea why this question get so horribly down voted. I suspect it's due to angry Theravadins who see this as being against Theravada orthodoxy of final permanent Nibanna.

The fact is the Lotus Sutra aims at deliberately over turning such conception about Arahatship, Nirvana and Buddhahood. Mahayana Buddhism relies on Prajnaparamita sutras as to establish that there is no final permanent nirvana nor the fact that Arhats are merely to remain as Arhats, but should aspire to Buddhahood with the full capacity of a Buddha. The core doctrine used to explain this view is through the understanding of dependent generation, which consequentially result in the impermanence and hence emptiness and non dual nature of any phenomena.

In the Lotus Sutra, it is therefore posited that Sariputra (Sariputta) shall one day attain full and complete unsurpassed awakening as a completely realized Buddha. The sutra further establish that only with such view, would they be True Arhats.

The sutra further explained that it is through skillful means that the Buddha teach sentient beings to give up their desires and attain Nirvana. Because most beings want to end their suffering but are afraid of taking on the challenge to become a Buddha. This is compared to the story of a rich man whose son have ran away from home, and now becomes a poor labourer. Now the poor man eventually reunite but no longer recognize his own father and is afraid of him thinking he's unworthy of being equal to the rich man. The rich man assigned the poor son various tasks until he gained his confidence and can truly inherit his father's legacy. The rich man is the Buddha who assigned the poor man that is sentient beings work, I.e. the way to end suffering Nibanna. But they should not be content with merely Nirvana but should aspire to be truly equal with the rich man e.g. the Buddha.