Enlightment of Early Monks

Many monks & nuns attained Arahantship, being released even in both ways;

When craving, Vaccha, has been got rid of by a monk, cut off at the root, made like a palm-tree stump that can come to no further existence in the future, he is a monk who is a perfected one, canker-waned, who has lived the life, done what was to be done, laid down the burden, attained his own goal, the fetters of becoming quite destroyed, freed by right profound knowledge.”

“Let be the good Gotama. Has the good Gotama even one monk who is a disciple and who, by the destruction of the cankers, having realised here and now by his own super-knowledge the freedom of mind and the freedom through intuitive wisdom that are cankerless, entering on them is abiding in them?” “Not merely a hundred, Vaccha, nor two hundred, three hundred, four hundred nor five hundred, but far more are those monks, disciples of mine, who, by the destruction of the cankers, having realised here and now by their own super-knowledge the freedom of mind and the freedom through intuitive wisdom that are cankerless, entering on them are abiding in them.”

“Let be the good Gotama, let be the monks. But has the good Gotama even one nun who is a disciple and who, by the destruction of the cankers, having realised here and now by her own super-knowledge the freedom of mind and the freedom through intuitive wisdom that are cankerless, entering on them is abiding in them?” “Not merely a hundred, Vaccha, nor two, three, four or five hundred, but far more are those nuns, disciples of mine, who, by the destruction of the cankers, having here and now realised by their own super-knowledge the freedom of mind and the freedom through intuitive wisdom that are cankerless, entering on them are abiding in them.” -mn73

As to laymen & lay-women there were none that attained the final goal at the breakup of the body without abandoning the fetters of householdership;

“Good Gotama, is there any householder who, not getting rid of the householder's fetter(s), at the breaking up of the body is an end-maker of ill?” “There is not any householder, Vaccha, who, not getting rid of the householder's fetter(s), at the breaking up of the body is an end-maker of ill.” -mn71

Yes. Many became arahants (fully enlightened ones) in the Buddha's lifetime.

While the Buddha was living, Venerables Sariputta, Moggallana, Mahakaccana, Mahakassapa, Punna, Anuruddha, Upali, Rahula, Khema, Dhammadinna, Bahiya, Patacara, Angulimala, Bhaddiya, Sona, Khemaka, Sunita, Sumana, Vimala, Canda, Kisagotami, Revata, Subhuti, Anna Kondanna, Dabba Mallaputta, Ambapali, Subha etc. became arahants.

Ven. Ananda became an arahant after the Buddha's passing away.

You can search for these names in the Palikanon's Buddhist Dictionary of Pali Proper Names and in AccessToInsight's Index of Proper Names.

You can find sources for these, in various suttas, including in the Theragatha (stories of the elder monks) and Therigatha (stories of the elder nuns).

From "Arahants, Bodhisattvas, and Buddhas" by Ven. Bodhi:

Although the Buddha was the pioneer in discovering the path to liberation, this does not mean that his arahant disciples just selfishly reaped the benefits of the path and did nothing for others. To the contrary, in the suttas we can see that many of them became great teachers in their own right who were capable of guiding others towards liberation. The best known among them are Sāriputta, Mahākaccāna, Moggāllana, and Ānanda. There was the monk Puṇṇa who went to the barbarian Sunāparanta country, risking his life to teach the Dhamma to the people there. There were such nuns as Khemā and Dhammadinnā, who were outstanding preachers, Paṭācārā, who was a master of the discipline, and many others. For four hundred years, the Buddhist texts were preserved orally, transmitted from teachers to pupils, and obviously there had to be thousands of monks and nuns who dedicated their lives to learning the texts and teaching them to pupils, all for the purpose of preserving the good Dhamma and Vinaya in the world.

The example established by the Buddha's great arahant disciples has been the model for the followers of the arahant ideal throughout history. While those who pursue this ideal do not make such lofty vows as do followers of the bodhisattva ideal, they are inspired by the example of the Buddha and his great disciples to work for the spiritual and moral uplift of others to the best of their ability: by teaching, by example, and by direct spiritual influence, inspired by the Buddha's command to "wander forth for the welfare of the multitude, for the happiness of the multitude, out of compassion for the world, for the good, welfare, and happiness of devas and human beings."