HSA is not allowed even for people who are enrolled only in Medicare Part A, which is free.  It's rare for someone who turns 65 not to sign up for Part A, although many people defer enrolling in Part B because they have equivalent coverage from an employer plan.  

@GRRA What's your source for "IRS includes..."?

The source is probably IRS Pub 969, which doesn't explain (because it's written by people too young to worry about Medicare) that "enrolled in Medicare" means any part of Medicare -- not just B, but A.  And the Code section itself doesn't state "enrolled," it states "eligible" and "entitled:"

Section 223(b) (7) Medicare eligible individuals

@BobKamman Thanks. That's what I thought.

@GRRA IRS Publications are not authoritative. As illustrated here sometimes they are wrong.

IRS Notice 2004-50:

Q-2. May an otherwise eligible individual who is eligible for Medicare, but not enrolled in Medicare Part A or Part B, contribute to an HSA?

A-2. Yes. Section 223(b)(7) states that an individual ceases to be an eligible individual starting with the month he or she is entitled to benefits under Medicare. Under this provision, mere eligibility for Medicare does not make an individual ineligible to contribute to an HSA. Rather, the term “entitled to benefits under” Medicare means both eligibility and enrollment in Medicare. Thus, an otherwise eligible individual under section 223(c)(1) who is not actually enrolled in Medicare Part A or Part B may contribute to an HSA until the month that individual is enrolled in Medicare.

Example (1). Y, age 66, is covered under her employer’s HDHP. Although Y is eligible for Medicare, Y is not actually entitled to Medicare because she did not apply for benefits under Medicare (i.e., enroll in Medicare Part A or Part B). If Y is otherwise an eligible individual under section 223(c)(1), she may contribute to an HSA.

Example (2). In August 2004, X attains age 65 and applies for and begins receiving Social Security benefits. X is automatically enrolled in Medicare. As of August 1, 2004, X is no longer an eligible individual and may not contribute to an HSA.

https://www.irs.gov/irb/2004-33_IRB#NOT-2004-50

I learn something new every day.