diff --git a/doc/source/whatsnew/v1.1.0.rst b/doc/source/whatsnew/v1.1.0.rst
index 10522ff797c59..6a6c7ebd49db1 100644
--- a/doc/source/whatsnew/v1.1.0.rst
+++ b/doc/source/whatsnew/v1.1.0.rst
@@ -843,6 +843,9 @@ Datetimelike
 - Bug in :meth:`DatetimeIndex.intersection` and :meth:`TimedeltaIndex.intersection` with results not having the correct ``name`` attribute (:issue:`33904`)
 - Bug in :meth:`DatetimeArray.__setitem__`, :meth:`TimedeltaArray.__setitem__`, :meth:`PeriodArray.__setitem__` incorrectly allowing values with ``int64`` dtype to be silently cast (:issue:`33717`)
 - Bug in subtracting :class:`TimedeltaIndex` from :class:`Period` incorrectly raising ``TypeError`` in some cases where it should succeed and ``IncompatibleFrequency`` in some cases where it should raise ``TypeError`` (:issue:`33883`)
+- Bug in constructing a Series or Index from a read-only NumPy array with non-ns
+  resolution which converted to object dtype instead of coercing to ``datetime64[ns]``
+  dtype when within the timestamp bounds (:issue:`34843`).
 - The ``freq`` keyword in :class:`Period`, :func:`date_range`, :func:`period_range`, :func:`pd.tseries.frequencies.to_offset` no longer allows tuples, pass as string instead (:issue:`34703`)
 
 Timedelta
diff --git a/pandas/_libs/tslibs/conversion.pyx b/pandas/_libs/tslibs/conversion.pyx
index 40b2d44235d8b..0811ba22977fd 100644
--- a/pandas/_libs/tslibs/conversion.pyx
+++ b/pandas/_libs/tslibs/conversion.pyx
@@ -167,7 +167,8 @@ def ensure_datetime64ns(arr: ndarray, copy: bool=True):
     """
     cdef:
         Py_ssize_t i, n = arr.size
-        int64_t[:] ivalues, iresult
+        const int64_t[:] ivalues
+        int64_t[:] iresult
         NPY_DATETIMEUNIT unit
         npy_datetimestruct dts
 
diff --git a/pandas/tests/base/test_constructors.py b/pandas/tests/base/test_constructors.py
index e27b5c307cd99..697364fc87175 100644
--- a/pandas/tests/base/test_constructors.py
+++ b/pandas/tests/base/test_constructors.py
@@ -13,6 +13,19 @@
 from pandas.core.base import NoNewAttributesMixin, PandasObject
 
 
+@pytest.fixture(
+    params=[
+        Series,
+        lambda x, **kwargs: DataFrame({"a": x}, **kwargs)["a"],
+        lambda x, **kwargs: DataFrame(x, **kwargs)[0],
+        Index,
+    ],
+    ids=["Series", "DataFrame-dict", "DataFrame-array", "Index"],
+)
+def constructor(request):
+    return request.param
+
+
 class TestPandasDelegate:
     class Delegator:
         _properties = ["foo"]
@@ -145,3 +158,14 @@ def test_constructor_datetime_outofbound(self, a, klass):
         msg = "Out of bounds"
         with pytest.raises(pd.errors.OutOfBoundsDatetime, match=msg):
             klass(a, dtype="datetime64[ns]")
+
+    def test_constructor_datetime_nonns(self, constructor):
+        arr = np.array(["2020-01-01T00:00:00.000000"], dtype="datetime64[us]")
+        expected = constructor(pd.to_datetime(["2020-01-01"]))
+        result = constructor(arr)
+        tm.assert_equal(result, expected)
+
+        # https://github.com/pandas-dev/pandas/issues/34843
+        arr.flags.writeable = False
+        result = constructor(arr)
+        tm.assert_equal(result, expected)