I'm so confused. :-(
As written is the sentence correct. (That should have been the question.)
Sorry for the confusion.
As others have already said, your sentence is correct, except for the "laying" --> "lying" part.
Now, I'll walk us through the confusion, and I hope I do so without making things worse.
The original sentence (corrected for lay/lie):
She picked up a stick that was lying at her feet.
Your question:
Is this sentence present progressive tense?
The first thing to note is this:
"Tense" is not a feature of sentences; it's a feature of verbs. A verb may have tense. But a sentence has verbs, and thus you often hear "present tense sentence", for example. But this is really only short for "The verb in this sentence is in the present tense."
There is a problem, when a sentence contains more than one verb - and these verbs are in different tenses. The sentence in question is an example. It contains two verbs:
"picked up" - which is past tense simple.
and
"was lying" - which is past tense progressive
Actually, these two verbs do not differ in "tense" but in "aspect", but that's another topic. I don't want to complicate things even further, so I'll just say that "past tense simple" and "past tense progressive" are both sub-types of the blanket category: "past tense".
Since there are two verbs in one sentence which are in different tenses, there is room for misunderstandings, when you ask which tense the sentence is.
Now, there is a loose convention to give priority to the verb of the
main clause. That is, in your sentence, "picked up". And this is why absitinvidia talked about "picked up" rather than "was lying".
So, to summarise the facts:
You have two verbs, one in past tense simple (picked up), and one in past tense progressive (was lying).
You can avoid a lot of confusion by only talking about tense in connection with verbs, not sentences. There is less room for misunderstanding if you ask:
Is "was lying" present progessive tense?
The answer to that is: you're right about progressive part, but wrong about the present part. It's past tense.