Maybe. Opinions differ and the evidence is inconsistent.
BBC News has recently produced a radio documentary about this, I've summarised it below (although it's worth listening to the audio, it's interesting).
At least some of the claimed instances are likely psychogenic or have 'normal' causes (eg food poisoning):
Some instances may be unrelated. "We had a bunch of military folk in
the Middle East who claimed to have this attack - turned out they had
food poisoning," says one former official. "We need to separate the
wheat from the chaff," reckons Mark Zaid, who says members of the
public, some with mental health issues, approach him claiming to
suffer from microwave attacks. One former official reckons around half
the cases reported by US officials are possibly linked to attacks by
an adversary. Others say the real number could be even smaller.
A December 2020 report from the US National Academies of Sciences concluded that evidence exists to support the existence of 'Havana Syndrome':
A December 2020 report by the US National Academies of Sciences was a
pivotal moment. Experts took evidence from scientists and clinicians
as well as eight victims. ... The panel looked at psychological and other causes, but
concluded that directed, high energy, pulsed microwaves were most
likely responsible for some of the cases, similar to the view of James
Lin, who gave evidence.
But the US State Department, which sponsored the report, considers it 'only a plausible hypothesis':
But even though the State Department
sponsored the study, it still considers the conclusion only a
plausible hypothesis and officials say they have not found further
evidence to support it.
New evidence is emerging:
The BBC has learnt that new evidence is arriving as data is collected
and analysed more systematically for the first time. Some of the cases
this year showed specific markers in the blood, indicating brain
injury. These markers fall away after a few days and previously too
much time had elapsed to spot them. But now that people are being
tested much more quickly after reporting symptoms, they have been seen
for the first time.
The answer may be complex and multi-faceted:
The debate remains divisive and it is possible the answer is complex.
There may be a core of real cases, while others have been folded into
the syndrome. Officials raise the possibility that the technology and
the intent might have changed over time, perhaps shifting to try and
unsettle the US. Some even worry one state may have piggy-backed on
another's activities. "We like a simple label diagnosis," argues
Professor Relman. "But sometimes it is tough to achieve. And when we
can't, we have to be very careful not to simply throw up our hands and
Source: ‘Havana syndrome ’ and the mystery of the microwaves, Gordon Corera, BBC News, 9 September 2021
See also: The mystery of Havana syndrome (27 mins audio, documentary podcast from BBC News), 9 September 2021