A jivana-mukta may or may not possess special outwardly abilities or physical appearance compared to an ordinary person, in accordance with the desire of the Supreme.
In the Bhagavad Gita, Krishna gave a simple definition of a liberated soul to Arjuna:
अर्जुन उवाच स्थितप्रज्ञस्य का भाषा समाधिस्थस्य केशव । स्थितधीः किं
प्रभाषेत किमासीत व्रजेत किम् ॥ ५४ ॥
Arjuna said: O Kṛṣṇa, what are the symptoms of one whose consciousness
is thus merged in transcendence? How does he speak, and what is his
language? How does he sit, and how does he walk?
श्रीभगवानुवाच प्रजहाति यदा कामान्सर्वान्पार्थ मनोगतान् । आत्मन्येवात्मना तुष्टः स्थितप्रज्ञस्तदोच्यते ॥ ५५ ॥
The Supreme Personality of Godhead said: O Pārtha, when a man gives up
all varieties of desire for sense gratification, which arise from
mental concoction, and when his mind, thus purified, finds
satisfaction in the self alone, then he is said to be in pure
transcendental consciousness.
Jada Bharata was in constant meditation on Krishna and was thus protected from the thieves by Goddess Kālī.
Shrila Sukadeva Goswami makes the following statement regarding the jivana-mukta Jada-Bharata in the Bhagavata Purana:
न वा एतद्विष्णुदत्त महदद्भुतं यदसम्भ्रम: स्वशिरश्छेदन आपतितेऽपि
विमुक्तदेहाद्यात्मभावसुदृढहृदयग्रन्थीनां सर्वसत्त्वसुहृदात्मनां
निर्वैराणां साक्षाद्भगवतानिमिषारिवरायुधेनाप्रमत्तेन तैस्तैर्भावै:
परिरक्ष्यमाणानां तत्पादमूलमकुतश्चिद्भयमुपसृतानां भागवतपरमहंसानाम् ॥
२० ॥
Śukadeva Gosvāmī then said to Mahārāja Parīkṣit: O Viṣṇudatta, those
who already know that the soul is separate from the body, who are
liberated from the invincible knot in the heart, who are always
engaged in welfare activities for all living entities and who never
contemplate harming anyone are always protected by the Supreme
Personality of Godhead, who carries His disc [the Sudarśana cakra] and
acts as supreme time to kill the demons and protect His devotees. The
devotees always take shelter at the lotus feet of the Lord. Therefore
at all times, even if threatened by decapitation, they remain
unagitated. For them, this is not at all wonderful (SB 5.9.20)
This also applies to Prahlada and all other liberated souls described in the shastras.
Some jivana-muktas may have a rather unorthodox outwardly appearance. For example, Lord Rishabhadeva (a plenary expansion of Lord Vishnu) assumed the activities of an avadhuta and did not bother to protect his physical body during a forest fire.
तस्य ह वा एवं मुक्तलिङ्गस्य भगवत ऋषभस्य योगमायावासनया देह इमां
जगतीमभिमानाभासेन सङ्क्रममाण: कोङ्कवेङ्ककुटकान्दक्षिणकर्णाटकान्देशान्
यदृच्छयोपगत: कुटकाचलोपवन आस्यकृताश्मकवल उन्माद इव मुक्तमूर्धजोऽसंवीत
एव विचचार ॥ ७ ॥
Actually Lord Ṛṣabhadeva had no material body, but due to yoga-māyā He
considered His body material, and therefore, because He played like an
ordinary human being, He gave up the mentality of identifying with it.
Following this principle, He began to wander all over the world. While
traveling, He came to the province of Karṇāṭa in South India and
passed through Koṅka, Veṅka and Kuṭaka. He had no plan to travel this
way, but He arrived near Kuṭakācala and entered a forest there. He
placed stones within His mouth and began to wander through the forest,
naked and with His hair disheveled like a madman.
अथ समीरवेगविधूतवेणुविकर्षणजातोग्रदावानलस्तद्वनमालेलिहान: सह तेन ददाह ॥
८ ॥
While He was wandering about, a wild forest fire began. This fire was
caused by the friction of bamboos, which were being blown by the wind.
In that fire, the entire forest near Kuṭakācala and the body of Lord
Ṛṣabhadeva were burnt to ashes (SB 5.6.7-8)
Lord Hanuman on the other hand exhibited divine opulences (ashta-siddhis) for the service of Sri Ram, which was in accordance with His Lord's desire.
Similarly, the sage Valmiki upon self-realization became a trikala-darsi and composer of Ramayana.