Don't know if you already tried this but I might suggest that you take cuttings from NEW wood.

Not sure about Lantana but Hydrangeas root very easily by taking cuttings from the new growth in summer.

Yes, they root very easily in new wood in rooting hormone in moist sterile potting soil. I would try late spring and maybe again in early summer. Also, my experience has been that for cuttings to survive outside, they need to have old wood because they die back to the old wood. So, you might need to over-winter your cuttings the first winter in pots and then plant them outside the following spring.

Lantana is not reliably hardy in z7. I also overwintered one, and have had some luck with cuttings- of the bright green new stems, not the brown stems. If you are keeping it trimmed as a topiary, you might not have many green stems long enough for a cutting.

My experience Lantana is very easy to root. I often just nip off new tips and stick them right in the ground. By summer/fall, it is a full grown plant. I love pure white lantana and got most of what I have just by doing that.
I have always wanted to create a standard lantana and have my instructions in my file but just haven't gotten around to doing it yet. That plant would have to be brought in the garage overwinter though (I think). What color is your topiary? I would love to see a picture of it.
Pauline in Wilmington NC

Can you cut off part of a branch that is brown and plant it directly into the ground? Will it take root?

Thanks

Take the green soft cuttings in summer when the soil temps remain high. If you wait until fall (l leave my cuttings outside - more humidity) they may rot.

Lantanas are weeds here, but nothing is a more reliable bloomer.