No, the liturgical books do not permit lay people or deacons to have open/extended hands during the Our Father.
From the General Instruction of the Roman Missal (GIRM):
“237. Then the principle celebrant, with hands joined, says the introduction to the Lord’s Prayer. Next, with hands extended, he says the Lord’s Prayer itself with the other concelebrants, who also pray with hands extended, and together with the people.”
From the Ceremonial of Bishops, Stational Mass of the Diocesan Bishop:
“159 After the doxology of the eucharistic prayer, the bishop, with hands joined, introduces the Lord’s Prayer, which all then sing or say; the bishop and the concelebrants hold their hands outstretched.”
From the GIRM: “124. … When the Entrance Chant is concluded, with everybody standing, the Priest and faithful sign themselves with the Sign of the Cross.”
Chapter 4 of the Ceremonial of Bishops is titled “General Norms”. It has near the beginning of the chapter “Before describing the individual rites, it seems advisable to state some general norms that have proved valid by long use and that should be followed”. Section V is headed “POSITION OF THE HANDS”. It begins:
“Raised and outstreched hands 104. Customarily in the Church a bishop or presbyter addresses prayers to God while standing and with hands slightly raised and outstreched. …”
Another part of this is: “Joined hands 107. Unless the bishop is holding the pastoral staff, he keeps his hands joined: [footnote 80. “Hands joined” means: “Holding the palms sideward and together before the breast, with the right thumb crossed over the left” (Caeremoniale Episcoporum, ed. 1886, I, XIX, 1) when, vested, he walks in procession for the celebration of a liturgy; when he is kneeling at prayer; when he moves from altar to chair or from chair to altar; when the liturgical books prescribe joined hands. Similarly, concelebrants and ministers keep their hands joined when walking from place to place or when standing, unless they are holding something.”
I also discuss this issue in a video at https://www.youtube.com/watch?v=ZktvVhOFyF0 which is about 6 minutes.