reported speech backshift
Yes. Here are a few examples I found on the Corpus of Contemporary American English searching for "told her I have." The first and third example feature present perfect; the second example features present tense:
She asked me if I've changed my diet and exercise routine over the past few years. I told her I haven't. ("Exercise and Eat Well: How Doctors Dole Out Advice," Talk of the Nation, 6 September 2012.)
Toward the end of that first meeting, she asked about my family. I told her I have a 22-year-old daughter. (Dennis Gersten, "Holy Madness in Healing." Psychology Today, 31.2, 1998, p. 59.)
But later on when Douglas said, " I didn't know you'd told your mother about the abortion, " Cara found herself replying protectively, " I didn't. I only told her I haven't been feeling so well. [...]" (Abby Frucht, Are You Mine? Grove Press, 1993.)
According to EF, "Tense Changes When Using Reported Speech," while past perfect is often used in reported speech to represent past and perfect verbs, reported speech may leave the tense unchanged in two situations:
- if the reporting verb is in present tense (e.g. "She asks," rather than "she asked")
- if the original statement is about something that is still true
In other words, if the information is still current in that context, you are allowed to leave the tense unchanged. In all three of the examples above, the speaker refers to something that is still true:
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The speaker has still not changed their diet and exercise routine.
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The speaker still has a 22 year old daughter.
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The speaker is still feigning to her mother that she hasn't been feeling well.
So, for your example, if it's clear that what you're saying is still true, the present perfect or the past perfect form may be used. Many times, this comes down to speaker choice. For example, is this assignment a current one or a long-past one?
Earlier today, Dorothy asked how far along I was on the current assignment. I told her I have already finished my essay.
At some point last year, Dorothy asked how much schoolwork I had. I told her I had already finished my essay.