diff --git a/CHANGELOG-unreleased.md b/CHANGELOG-unreleased.md
index 6d7dae96c..9b2eb290f 100644
--- a/CHANGELOG-unreleased.md
+++ b/CHANGELOG-unreleased.md
@@ -15,6 +15,8 @@ the released changes.
 - `ssb_to_psb_ICRS` implementation is now a lot faster (uses erfa functions directly)
 - `ssb_to_psb_ECL` implementation within `AstrometryEcliptic` model is now a lot faster (uses erfa functions directly)
 - Upgraded versioneer for compatibility with Python 3.12
+- Creation of `Fitter` objects will fail if there are free unfittable parameters in the timing model.
+- Only fittable parameters will be listed as check boxes in the `plk` interface.
 ### Added
 - CHI2, CHI2R, TRES, DMRES now in postfit par files
 - Added `WaveX` model as a `DelayComponent` with Fourier amplitudes as fitted parameters
@@ -24,6 +26,7 @@ the released changes.
 - Support for wideband data in `pint.bayesian` (no correlated noise).
 - Added `DMWaveX` model (Fourier representation of DM noise)
 - Piecewise orbital model (`BinaryBTPiecewise`)
+- `TimingModel.fittable_params` property
 - Simulate correlated noise using `pint.simulation` (also available via the `zima` script)
 ### Fixed
 - Wave model `validate()` can correctly use PEPOCH to assign WAVEEPOCH parameter
@@ -33,4 +36,5 @@ the released changes.
 - Fixed an incorrect docstring in `pbprime()` functions. 
 - Fix ICRS -> ECL conversion when parameter uncertainties are not set.
 - `get_TOAs` raises an exception upon finding mixed narrowband and wideband TOAs in a tim file. `TOAs.is_wideband` returns True only if *ALL* TOAs have the -pp_dm flag.
+- `TimingModel.designmatrix()` method will fail with an informative error message if there are free unfittable parameters in the timing model.
 ### Removed
diff --git a/src/pint/fitter.py b/src/pint/fitter.py
index 2e6a31a9b..ce678edc2 100644
--- a/src/pint/fitter.py
+++ b/src/pint/fitter.py
@@ -217,6 +217,16 @@ class Fitter:
     """
 
     def __init__(self, toas, model, track_mode=None, residuals=None):
+        if not set(model.free_params).issubset(model.fittable_params):
+            free_unfittable_params = set(model.free_params).difference(
+                model.fittable_params
+            )
+            raise ValueError(
+                f"Cannot create fitter because the following unfittable parameters "
+                f"were found unfrozen in the model: {free_unfittable_params}. "
+                f"Freeze these parameters before creating the fitter."
+            )
+
         self.toas = toas
         self.model_init = model
         self.track_mode = track_mode
diff --git a/src/pint/models/binary_ell1.py b/src/pint/models/binary_ell1.py
index 8e3c93b4e..01b619c0e 100644
--- a/src/pint/models/binary_ell1.py
+++ b/src/pint/models/binary_ell1.py
@@ -433,6 +433,11 @@ def __init__(self):
         self.binary_model_name = "ELL1k"
         self.binary_model_class = ELL1kmodel
 
+        self.remove_param("OMDOT")
+        self.remove_param("EDOT")
+        self.remove_param("EPS1DOT")
+        self.remove_param("EPS2DOT")
+
         self.add_param(
             floatParameter(
                 name="OMDOT",
@@ -451,9 +456,6 @@ def __init__(self):
             )
         )
 
-        self.remove_param("EPS1DOT")
-        self.remove_param("EPS2DOT")
-
     def validate(self):
         """Validate parameters."""
         super().validate()
diff --git a/src/pint/models/timing_model.py b/src/pint/models/timing_model.py
index 113ff7fa1..0b6350548 100644
--- a/src/pint/models/timing_model.py
+++ b/src/pint/models/timing_model.py
@@ -53,6 +53,7 @@
     Parameter,
     boolParameter,
     floatParameter,
+    funcParameter,
     intParameter,
     maskParameter,
     strParameter,
@@ -592,7 +593,8 @@ def params(self):
 
     @property_exists
     def free_params(self):
-        """List of all the free parameters in the timing model. Can be set to change which are free.
+        """List of all the free parameters in the timing model.
+        Can be set to change which are free.
 
         These are ordered as ``self.params`` does.
 
@@ -615,6 +617,17 @@ def free_params(self, params):
                 f"Parameter(s) are familiar but not in the model: {params}"
             )
 
+    @property_exists
+    def fittable_params(self):
+        """List of parameters that are fittable, i.e., the parameters
+        which have a derivative implemented. These derivatives are usually
+        accessed via the `d_delay_d_param` and `d_phase_d_param` methods."""
+        return [
+            p
+            for p in self.params
+            if (p in self.phase_deriv_funcs or p in self.delay_deriv_funcs)
+        ]
+
     def match_param_aliases(self, alias):
         """Return PINT parameter name corresponding to this alias.
 
@@ -1965,16 +1978,38 @@ def designmatrix(self, toas, acc_delay=None, incfrozen=False, incoffset=True):
         units : astropy.units.Unit
             The units of the corresponding parts of the design matrix
 
-        Note
-        ----
-        Here we have negative sign here. Since in pulsar timing
-        the residuals are calculated as (Phase - int(Phase)), which is different
-        from the conventional definition of least square definition (Data - model)
-        We decide to add minus sign here in the design matrix, so the fitter
-        keeps the conventional way.
+        Notes
+        -----
+        1. We have negative sign here. Since the residuals are calculated as
+        (Phase - int(Phase)) in pulsar timing, which is different from the conventional
+        definition of least square definition (Data - model), we have decided to add
+        a minus sign here in the design matrix so that the fitter keeps the conventional
+        sign.
+
+        2. Design matrix entries can be computed only for parameters for which the
+        derivatives are implemented. If a parameter without a derivative is unfrozen
+        while calling this method, it will raise an informative error, except in the
+        case of unfrozen noise parameters, which are simply ignored.
         """
 
         noise_params = self.get_params_of_component_type("NoiseComponent")
+
+        if (
+            not set(self.free_params)
+            .difference(noise_params)
+            .issubset(self.fittable_params)
+        ):
+            free_unfittable_params = (
+                set(self.free_params)
+                .difference(noise_params)
+                .difference(self.fittable_params)
+            )
+            raise ValueError(
+                f"Cannot compute the design matrix because the following unfittable parameters "
+                f"were found unfrozen in the model: {free_unfittable_params}. "
+                f"Freeze these parameters before computing the design matrix."
+            )
+
         # unfrozen_noise_params = [
         #     param for param in noise_params if not getattr(self, param).frozen
         # ]
@@ -2918,7 +2953,11 @@ def __init__(self):
     def __repr__(self):
         return "{}(\n    {})".format(
             self.__class__.__name__,
-            ",\n    ".join(str(getattr(self, p)) for p in self.params),
+            ",\n    ".join(
+                str(getattr(self, p))
+                for p in self.params
+                if not isinstance(p, funcParameter)
+            ),
         )
 
     def setup(self):
diff --git a/src/pint/pintk/plk.py b/src/pint/pintk/plk.py
index 368e19599..3a32ad013 100644
--- a/src/pint/pintk/plk.py
+++ b/src/pint/pintk/plk.py
@@ -208,6 +208,7 @@ def addFitCheckBoxes(self, model):
                 for p in model.components[comp].params
                 if p not in pulsar.nofitboxpars
                 and getattr(model, p).quantity is not None
+                and p in model.fittable_params
             ]
 
             # Don't bother showing components without any fittable parameters