Yes, Lord Dattatreya did have a wife. She is mentioned in Chapter 16 of the Markandeya Purana.
Dattatreya sported with his wife and drank liquor with her in front of the Munis to test their loyalty to him. Inspite of all his acts, the Munis refused to leave him.
मुनिपुत्रवृतो योगी दत्तात्रेयोऽप्यसङ्गिताम्।। १०८॥
अभीप्समानः सरसि निममज्ज चिरं विभुः।
तथापि तं महात्मानमतीव प्रियदर्शनम्॥ १०९॥
तत्यजुर्न कुमारास्ते सरसस्तीरसंश्रयाः।
Surrounded by the sons of Munis, the lordly yogi Dattatreya also,
desirous of obtaining exemption from all attachments, long immersed
himself in a lake. Nevertheless those youths, resorting to the bank of
the lake, did not forsake him, who was magnanimous and exceedingly
benign.
दिव्ये वर्षशते पूर्णे यदा तेन त्यजन्ति तम्।। ११०॥
तत्प्रीत्या सरसस्तीरं सर्वे मुनिकुमारकाः। ततो दिव्याम्बरधरां सुरूपां
सुनितम्बिनीम्।। १११॥
नारीमादाय कल्याणीमुक्ततार जलान्मुनिः।
स्त्रीसन्निकर्षिणं ह्येते परित्यक्ष्यन्ति मामिति॥ ११२॥
मुनिपुत्रास्ततो योगे स्थास्यामीति विचिन्तयन्।
When after a hundred heavenly years were ended, all those youthful
Munis, through affection for him, still forsook not the bank of the
lake, the Muni, taking his noble wife clothed in heavenly raiment,
beautiful and plump in form, arose from the water, thinking, If these
sons of Munis shall forsake me because of the presence of a woman,
then I will remain free from all attachments.
तथापि ते मुनिसुता न त्यजन्ति यदा मुनिम्।। ११३।।
ततः सह तया नार्या मद्यपानमथाकरोत्। सुरापानरतं तेन सभार्यं
तत्यजुस्ततः।। ११४।।
गीतवाद्यादिवनिताभोगसंसर्गदूषितम्। मन्यमाना महात्मानं तया सह
बहिष्क्रियम्॥ ११५।।
When nevertheless the sons of the Munis did not forsake him, he next
drank intoxicating liquors in company with his wife. Thereupon they
did not forsake him, though he was engrossed in drinking spirituous
liquor in company with his wife, and though he was rendered impure by
addiction to singing, musical instruments and Such like, and also by
intercourse with his wife; deeming that the high-souled Muni when with
her was detached from religious rites.
नावाप दोष योगीशो वारुणीं स पिबन्नपि। अन्तावसायिवेश्मान्तर्मातरिश्वा
स्पृशनिव। ११६।।
सुरां पिबन्सपत्नीकस्तपस्तेपे स योगवित्। योगीश्वरश्चिन्त्यमानो
येागिभिर्मुक्तिकांक्षिभिः।। ११७।।
The lord of yogis, although drinking spirituous liquor, incurred no
fault. Dwelling like Matarisvan within the abodes of candalas,
drinking strong drink he, skilled in yoga, the lord of yogis,
attended by his wife, performed austerities, being meditated on by
yogis who longed for deliverance from mundane existence.
Further on in the same chapter Sage Garga asks Kartavirya Arjuna to pray to Dattatreya citing an old story where the Danavas kidnapped Dattareya's wife and met their destruction. Here her name is given as Laksmi.
When the Daityas overpowered the gods Brihaspati advises the gods to pray to Dattatreya and ask for a boon that would destroy the Daityas
बृहपतिरुवाच
दत्तात्रेयं महाभागमत्रेः पुत्रं तपोधनम्। विकृताचरण भक्त्या
सन्तोषयितुमर्हथ। १४०॥
स वो दैत्य विनाशाय वरदो दास्यते वरम्। ततो हनिष्यथ सुराः
सहितान्दैत्यदानवान्॥१४१॥
Brihaspati said- "Deign to gratify with your faith Dattātreya, Atri's
high-souled son, the ascetic, who is occupied in improper practices.
He the boongiver will grant you a boon for the destruction of the
Daityas; then, O gods, shall you and your friends slay the Daityas and
Danavas."
हन्तुं शक्ता न सन्देहो दत्तात्रेयप्रसादतः। इत्युक्तास्ते तदा
जग्मुर्दत्तात्रेयाश्रमं सुराः॥ १४२॥
ददृशुश्च महात्मानं क्षान्तं लक्ष्म्या समन्वितम्। उद्गीयमानं गन्धर्वैः
सुरापानरतं मुनिम्॥ १४३।।
Thus exhorted the gods then went to Dattatreya's hermitage, and they
beheld the highsouled Muni, attended by Laksmi, hymned by
Gandharvas, and engrossed in quaffing spirituous liquor.
गर्ग उवाच एवमुक्तस्ततो देवैर्दत्तात्रेयोऽब्रवीदिदम्। प्रहस्य
त्रिदशान्सर्वान्यद्येतद्भवतां मतम्॥१५५॥
तदाहूयासुरान्सर्वान्युद्धाय सुरसत्तमाः। इहानयत मद् दृष्टिगोचरं मा
विलम्ब्यताम्॥१५६॥
मददुष्टिपातहुतभुक्प्रक्षीणबलतेजस:। येन नाशमशेषास्ते प्रयान्ति मम
दर्शनात्॥ १५७।।
Garga spoke
Thus accosted by the gods, Dattatreya then with a smile spoke thus to
all the thirty gods;- "If this be your opinion, then summon all the
Asuras to battle, O most virtuous gods and bring them here before my
view-delay you not-in order that the glory of their strength may be
consumed by the fire of my glance, and that they may all perish from
my sight.
The Daityas break into Dattatreya's hermitage and resolve to carry off Laksmi. But as they carried her on their heads she deserts them and the sin of her kidnapping is attached to them and they lose their merits. So the Gods defeat them easily and Laksmi returns to Dattatreya.
ददृशुस्तं महात्मानं दत्तात्रेयं मदालसम्। १६०॥
वामपार्श्वस्थितामिष्टामशेषजगतः शुभाम्। भार्या चास्य सुचार्वङ्गीं
लक्ष्मीमिन्दुनिभाननाम्।। १६१॥
नीलोत्पलाभनयनां पीनश्रोणिपयोधराम्। सुदतीं मधुराभाषां
सर्वयोषिद्बुणैर्युताम्।। १६२।।
Even there the Daityas penetrated, driving forward the
heaven-dwellers, and saw the highsouled mighty Dattātreya; and
seated at his left side Laksmi, loved by all the worlds beauteous, her
shape most graceful, her countenance like the moon, her eyes lustrous
as the blue water-lily, her hips large and breasts full, uttering
melodious speech, adorned with every womanly virtue.
दृष्ट्वाग्रतस्तदा दैत्याः साभिलाषमनोभवाः। न शेकुरुद्धता दैत्या मनस
बोखुमातुराः॥१६३॥
त्यक्ता देवान्स्त्रियं तां तु हर्तुकामा हतौजसः। प्रेरितास्तेन पापेन
ह्यासक्तास्ते ततोऽब्रुवन्॥ १६४॥
Seeing her before them, the Daityas seized with longing, could not
bear the intense love with fortitude, and pined in mind to carry her
off. Desisting from the gods, but desirous of seizing the lady, they
were shattered in vigour, being bewitched by that sin.
गर्ग उवाच
सानुरागास्ततस्ते तु मुनेरन्तिकमागमन्। तस्य तां योषितं साध्वीं
समुत्क्षिप्य स्मरातुराः।। १६७॥
शिबिकायां समारोष्य सहिता दैत्यदानवा:। शिरः सुशिबिकां कृत्वा
स्वस्थानाभिमुखा ययुः।। १६८ ।।
Thereupon possessed with longing and thus mutually exhorted afflicted
by love, the united Daityas and Danavas raised up his virtuous wife,
mounted her in the palki, and placing the palki on their heads set off
for their own homes.
दत्तात्रेयस्तथा देवान्विहस्येदमथाब्रवीत्।
दिष्ट्यां च हन्त दैत्यानामेषा लक्ष्मीः शिरोगता।
सप्तस्थानान्यतिक्रम्य लयमन्यमुपेष्पति।। १६९॥
Thereon Dattātreya smiling spoke thus to the gods- "Bravo! you
prosper! Here is Laksmi borne on the heads of the Daityas. She has
passed beyond the seven stations, she will reach another,
देवा ऊचुः
कथयस्व जगन्नाथ केषु स्थानेष्ववस्थिता। पुरुषस्य फलं किं वा प्रयच्छत्यथ
नश्यति।। १७०॥
The gods spoke
Say, O lord of the world, in what stations has she her abode; and what
result of a man's does she bestow or destroy?
मृष्टात्र वाक्यलावण्यमाज्ञामवितथां तथा। मुखस्थिता कवित्वं च
यच्छत्युदधिसम्भवा।। १७४॥
शिरोगता सन्त्यजति ततोऽन्यं याति चाश्रयम्। सेयं शिरोगता
दैत्यान्परित्यजति साम्प्रतम्॥ १७५॥
When abiding in the countenance, the sea-born goddess bestows beauty
fashioned according to her word, real command also, and poetic fire.
When mounted on the head, she forsakes the man and thence resorts to another abode. And here, mounted on their head, she will now desert
these Daityas,
प्रगृह्यास्त्राणि वध्यन्तां तस्मादेते सुरारयः। न भेतव्यं भृशं त्वेते
मया निस्तेजसः कृताः॥ १७६॥
परदारावमर्शाच्च दग्धपुण्या हतौजसः। तस्मादेतेऽभिहन्यन्तां
भवद्भिरविशंकितैः॥ १७७॥
Therefore seize your arms and slay these foes of the gods; nor fear
them greatly; I have rendered them impotent; and through touching
anothers wife their merit is consumed, their might is broken.
गर्ग उवाच
ततस्ते विविधैरस्त्रैर्वध्यमानाः सुरारयः। शिरःसु लक्ष्म्याप्याक्रान्ता
विनेशुरिति नः श्रुतम्॥ १७८॥
लक्ष्मीश्रोत्पत्य सम्प्राप्ता दत्तात्रेयं महामुनिम्। स्तूयमाना सुरैः
सेन्द्रैर्दैत्यनाशान्मुदान्वितैः॥ १७९॥
प्रणिपत्य ततो देवा दत्तात्रेयं महामुनिम्। जय कृष्ण जगन्नाथ दैत्यान्तक
हरप्रभो॥ १८०॥
नारायणाच्युतानन्त वासुदेवाक्षयाजर। त्वत्प्रसादात्सुखं लक्ष्मी राज्यं
सम्पज्जनार्दन॥ १८१॥
शार्ङ्गधन्वंश्चक्रपाणे भक्तानां नित्यवत्सल।। इति स्तुत्वा नाकपृष्ठं
यथापूर्वं गताः सुराः।। १८२॥
Garga spoke
Thereupon those enemies of the gods, being slain by divers weapons and
their heads being assailed by Laksmi, perished-thus have we heard.
And Laksmi, flying up, reached the great Muni Dattātreya, being hymned by all the gods who were filled with joy at the slaughter of
the Daityas Thereupon the gods, prostrating themselves before the
wise Dattātreya, gained as before the uppermost heaven, being freed
from affliction.