The short answer is no. The IRS explains in Publication 526 that "You cannot deduct the value of your time or services, including... [t]he value of income lost while you work as an unpaid volunteer for a qualified organization."
However, there's an alternative strategy that freelancers might consider. The freelancer could charge the non-profit for their services, and then turn around and make a tax-deductible contribution for the same amount. This strategy has multiple tax consequences, so let's see how the math would work out in one scenario.
Let's assume, for our example, that an freelancer is filing a Schedule C to report her income, and she also itemizes her deductions, and she is in the 25% tax bracket. She charges the non-profit $1,000 for some web design work, and then separately she donates $1,000 to the charity. In this scenario, our freelancer has increased her self-employment income by $1,000, and so this extra income will increase her self-employment tax by roughly $141. This transaction would increase her deduction for half of the self-employment tax by roughly $70, and would increase her charitable deductions by $1,000. Now her income has increased $1,000, but this will be offset by the charity deduction. So overall, her taxable income is reduced only by the extra $70 deduction for self-employment tax, which would lower her income tax by about $17. Her tax impact: she pays an extra $141 of self-employment tax, but pays $17 less in income taxes, leaving her with $124 in higher taxes overall.
In this scenario, the freelancer does not come out ahead. The charity effectively pays nothing for the services, but the freelancer generates a tax hike for herself because the increase in her self-employment taxes is larger than the tax savings. The freelancer would be better off volunteering her services for free rather than charging a fee, and then turning around and donating the same amount.
However, this scenario may not fit your exact situation. So it would make sense to run your own numbers to see how it would work on your tax return.